cambridge textbook

203

C h a p t e r

6Multistage events and

applications of probability

Syllabus topic PB2 Multistage events and applications of probability

In this chapter you will learn to:

Construct and use a tree diagram for a multistage event

Determine the number of outcomes for a multistage event

Determine the number of ordered and unordered selections

Use a probability tree to calculate the probability of events A and B

Use a probability tree to calculate the probability of event A or B

Calculate the expected outcomes of a simple experiment

Calculate financial expectation

6.1 Multistage eventsA multistage event consists of two or more events, for example, tossing two coins or selecting a card from a pack of cards and throwing a die. A tree diagram is often used to show all the possible outcomes or the sample space of a multistage event. It shows each event as a branch of the tree.

The tree diagram opposite shows all the possible outcomes for tossing two coins. The outcomes of the first event are listed (H or T) with 2 branches. The outcomes of the second event are listed (H or T) with 2 branches on each of the outcomes of the first event. The sample space is HH, HT, TH and TT.

Tree diagrams

Draw a tree diagram with each event as a new branch of the tree.

Always draw large clear tree diagrams and list the sample space on the right-hand side.

H

T

H

H

T

T

HH

HT

TH

TT

1st 2nd Sample space

6.1

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The Powers Family Trust 2013 Cambridge University Press

204 HSC Mathematics General 2

Probability is the chance of something happening. The event is denoted by the letter E and P(E) refers to the probability of event E. The probability of the event is calculated by dividing the number of favourable outcomes by the total number of outcomes.

Probability

En E

n SProbability (Event)

Number of favourable outcomes

Total number of outcomesor P( )

( )

( )= =

Example 1 Using a tree diagram for a multistage event

A coin is tossed and a die is rolled.a Construct a tree diagram of these two events showing the sample space.b What is the probability of throwing a tail and a 2?c What is the probability of a head and a number less than 4?

Solution 1 Draw the first branch for the first

event tossing a coin. 2 Tossing a coin has two outcomes

(Head or Tail) so there are two branches.

3 Draw the second branch for the second event rolling a die.

4 Rolling a die has six outcomes (1, 2, 3, 4, 5 or 6) so there are six branches. Draw 6 branches for each of the two outcomes from the first event.

5 Use the branches of the tree to list the sample space. Write the outcomes down the right-hand side (sample space).

6 Write the formula for probability. 7 Number of favourable outcomes (T2) is

1. The total number of outcomes is 12. 8 Substitute into the formula.

9 Write the formula for probability.10 Number of favourable outcomes

(H1, H2 or H3) is 3. The total number of outcomes is 12.

11 Substitute into the formula.12 Simplify the fraction.

a

H

T

Coin Die Sample space1

2

3

4

5

6

1

2

3

4

5

6

H1

H2

H3

H4

H5

H6

T1

T2

T3

T4

T5

T6

b P(T2)T2

=

=

n

n s

( )

( )1

12

c P(H1, H2, H3)H1, H2, H3)

=

=

=

n

n s

(

( )3

121

4



205Chapter 6 Multistage events and applications of probability

Exercise 6A 1 Two people are selected and their gender recorded.

a Find the sample space by completing the tree diagram.b What is the probability of obtaining two males?c What is the probability of obtaining a male and a

female in any order?

2 There are three questions in a Yes or No survey.a Find the sample space by completing the tree

diagram.b How many possible outcomes are there?c What is the probability of choosing, in order, a

No, Yes and Yes?d What is the probability of choosing three Yes

responses?

3 Two fair dice are thrown and the results recorded. Part of a tree diagram is shown below.

a What is the sample space?

c What is the probability of throwing a 4 with the first die?

e What is the probability of throwing a 1 then a 6?

b How many elements are in the sample space?

d What is the probability of throwing an odd number with the second die?

f What is the probability of throwing a 2 then a 3?

M

F

1st 2nd

1st 2nd 3rd

Y

N

1

2

3

4

5

6

1

2

3

4

5

6

11

12

13

14

15

16




4 Abbey and Oscar are planning to have two children.a Draw a tree diagram to find the sample space for the gender of their children.b What is the probability of having a boy then a girl?c What is the probability of having a girl then a boy?d What is the probability of having two girls?

5 A two-digit number is formed using the digits 1, 2 and 3. The same number cannot be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit.a Find the sample space from the tree diagram.b What is the probability that the number starts with a 3?c What is the probability that the number ends with a 1?d What is the probability that the number formed is 23?

6 A menu has two entrees (E1 and E2) and four mains (M1, M2, M3 and M4).a Construct a tree diagram to find the sample space.b What is the probability of choosing E1 for the entree?c What is the probability of choosing M4 for the mains?d What is the probability of choosing E1 and M3?e What is the probability of choosing E2 and either M1 or M2?

7 Three people (M, N and O) have applied for the supervisors position and two people (P and Q) have applied for a casual position. Assume all applicants have an equal chance of getting each position.a Construct a tree diagram to find the

sample space.b What is the probability that the

supervisor selected will be person M?

c What is the probability that the person selected for the casual position will be person Q?

d What is the probability of selecting person N as the supervisor and person P as the casual?

e What is the probability of selecting person M or person N then person P?

Tens Units

12

3

1

32

31

2




Development

8 William tosses a coin and spins a spinner that has white, pink and red sections. Use a tree diagram to find the sample space.

9 One cup contains two discs labelled A and B. A second cup contains two discs labelled C and D. A third cup contains two discs labelled E and F. A disc is chosen from each cup at random.a Use a tree diagram to find the sample space.b What is the probability of choosing A from the first cup?c Find the probability of choosing a C or D from the second cup?d What is the probability of choosing ABC?e What is the probability of choosing ACE or BDF?f What is the probability of choosing a disc labelled with a vowel?

10 A two-digit number is formed using the digits 1, 3 and 5. The same number can be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit. Use a tree diagram to find the sample space.

11 Three blue balls (B1, B2, B3) and one red ball (R1) are placed in a bag. Two balls are selected at random with replacement.a Use a tree diagram to determine the number of elements in the sample space.b What is the probability of selecting B1 twice?c What is the probability of selecting two blue balls?d What is the probability of selecting two red balls?e What is the probability of selecting B1 and R1?

12 Five cards (ace, king, queen, jack and ten) are placed face down on a table. One card is selected at random and the result recorded. This card is not returned to the table. A second card is then selected at random.a Use a tree diagram to determine the number of elements in the sample space.b What is the probability of selecting an ace followed by a jack?c What is the probability of choosing one king?

13 There are four candidates for the positions of principal and deputy principal. The four candidates are Amy, Barry, Chelsea and David. Construct a tree diagram with choosing the principal as the first event and choosing the deputy principal as the second event. Use a tree diagram to determine the sample space.




6.2 Number of arrangementsThe fundamental counting principle states that if we have p outcomes for first event and q outcomes for the second event, then the total number of outcomes for both events is p q. It simply involves multiplying the number of outcomes for each event together. Consider the multistage event of having two babies and the gender of each baby. The first baby has two outcomes (boy or girl) and the second baby has two outcomes (boy or girl). The total number of outcomes for both events is 2 2 = 4 (BB, BG, GB or GG). This principle can be extended to more than two events as shown in example 3.

Fundamental counting principle

Number of outcomes (two events) = p q

p Number of outcomes of the first event

q Number of outcomes of the second event

Example 2 Determining the number of arrangements

A coin is tossed and a die is rolled. How many different outcomes are possible?

Solution

1 The first event is tossing a coin. There are 2 possible outcomes (H or T).

2 The second event is rolling a die. There are 6 possible outcomes (1, 2, 3, 4, 5 and 6).

3 Multiply the number of outcomes for each event to determine the number of arrangements.

There are two different outcomes for the coin toss and six different outcomes for the die roll.

Number of outcomes = 6= 122


The first page of the Mathematics paper has three multiple-choice questions, each with four answers (A, B, C and D). How many different ways are there of answering these three questions?

Solution

1 The first event is the first question. There are4 possible outcomes (A, B, C and D).

2 The second event is the second question. There are 4 possible outcomes (A, B, C and D).

3 The third event is the third question. There are4 possible outcomes (A, B, C and D).


There are four outcomes for each of the three questions.

Number of outcomes = 4 = 644 4





Anthony, Bailey, Chloe and Donald are required to stand in a row for selection to a committee.a How many different arrangements are possible?b List all the possible outcomes.

A

B

B

C

D

DCDBCBDCDACADBDABABCCABA

CDBDBCCDADACBDADABCBACAB

C

D

A

C

D

A

B

D

A

B

C

Solution

1 The first event is the first person. There are 4 possible outcomes (A, B, C and D).

2 The second event is the second person. There are 3 possible outcomes.

3 The third event is the third person. There are 2 possible outcomes.

4 The fourth event is the fourth person. There is only 1 possible outcome.


6 Use the tree diagram or a table to list the 24 outcomes.

a Number of arrangements

= 3 2 1= 244

b Possible outcomes ABCD, ABDC, ACBD, ACDB, ADBC, ADCB BACD, BADC, BCAD, BCDA, BDAC, BDCA CABD, CADB, CBAD, CBDA, CDAB, CDBA DABC, DACB, DBAC, DBCA, DCAB, DCBA




Exercise 6B 1 Three picture cards (king, queen and jack) are placed in a row on the table.

a How many different arrangements are possible?b List all the possible arrangements.

2 Nathan places four different coloured pegs in a row on the table. The coloured pegs are blue, green, red and yellow.a How many different arrangements are possible?b List all the possible arrangements.

3 Max, Oliver, Noah and Peter are nominated for chairman and assistant chairman. What are all the possible combinations?

4 A fair die is tossed twice.a How many different outcomes are possible?b List all the possible outcomes.c If the die is tossed again, how many different outcomes are now possible?

5 The letters of the word PUNCHBOWL are to be rearranged.a How many different arrangements are possible?b How many different arrangements are possible if the letters PUNCH are removed?c How many different arrangements are possible if the letters BOWL are removed?

6 In how many ways can Kim, Molly, Nicky, Olivia and Paige stand in a queue?

7 Isabella has 4 hats, 5 shirts and 3 pairs of jeans.a How many different combinations are possible?b Isabella buys two more shirts. How many different combinations are now possible?

8 There are five parcels labelled D, E, F, G and H under the Christmas tree.a A parcel is chosen at random and removed from the tree. A second parcel is then

chosen and removed from the tree. How many different choices are possible?b A third parcel is then chosen and removed from the tree. How many different choices

for the three parcels are possible?




9 Ming has a four-digit PIN and he knows the digits are 2, 3, 4 and 5. However, he cannot remember the order of the digits.a How many different four-digit

PINs are possible?b What is the probability that

Ming will be able to correctly guess his PIN?

10 A swimming relay team consists of four swimmers chosen in order.a How many different teams are possible from six swimmers?b How many different teams are possible from seven swimmers?c How many different teams are possible from eight swimmers?

11 The letters from the word ASHBURY are being used to form other words.a How many two-letter arrangements are possible?b How many three-letter arrangements are possible?c How many four-letter arrangements are possible?

12 A refrigerator contains nine different cans of soft drink.a Emma chooses two cans of soft drink, the first for herself and the second for a friend.

How many different possible choices could Emma make?b Charlie selects two cans of soft drink after Emma has taken her two cans. How many

different possible choices could Charlie make?

13 Three cards labelled A, B and C can be arranged in any order.a In how many different ways can the cards be arranged?b What is the probability that the second card in an arrangement is a C?c What is the probability that the last card in an arrangement is not a C?

14 The letters of the word LAMBTON are to be rearranged.a How many different arrangements are possible?b What is the probability that the letter N will be the first letter?c What is the probability that the letters are arranged in alphabetical order

(ABLMNOT)?




Development

15 Four cards each with a different suit (diamond, heart, spade or club) are placed face down on the table. A card is selected from the table. Its suit is noted and the card replaced on the table. The cards are shuffled. A second card is then chosen and its suit noted.a How many possible outcomes are there?b What is the probability that both cards are diamonds?c What is the probability that both cards are not a diamond?d What is the probability that the first card is a club and the second card a heart?e What is the probability that both cards are black?f What is the probability that both cards are not black?

16 Lachlan is completing a true or false test on his laptop that has ten questions.a How many possible outcomes are

there?b What is the probability of

randomly getting every question correct?

c Lachlan is confident he has the first six questions correct. What is the probability of randomly getting the remaining questions correct?

17 Motor vehicle number plates consist of three letters followed by three digits.a How many different number plates are possible?b What is the probability that the number plate will be BON007?c What is the probability of the number plate starting with BON?d What is the probability of the number plate ending with 007?

18 A menu has four entrees (E1, E2, E3, E4), five mains (M1, M2, M3, M4, M5) and three desserts (D1, D2, D3).a How many different meals are possible?b How many different meals have M1 as the main meal?c What is the probability of choosing E1, M1 and D1?d What is the probability of choosing E2 as the entree and D2 as the dessert?e The restaurant has decided to remove one of the entrees from the menu. How many

different meals are now possible?




6.3 Ordered selectionsAn ordered selection or a permutation occurs when a selection is made from a group of items and the order is important. For example, drawing first, second and third prizes in a raffle or electing the captain and the vice-captain of the school. Consider the event of selecting two cards from four cards (1, 2, 3 and 4) to make a two-digit number. The order is critical. The first card is the tens digit (4 choices) and the second card is the units digit (3 choices). If the selection was a 2 followed by a 3 then the two-digit number is 23. However, if the selection was a 3 followed by a 2 then the two-digit number is 32. These are different outcomes. Order makes a difference to the result.

Ordered selection

Selection is made from a group of items and the order is important.

AB is different from BA.

Permutation nPr (n items available for selection and r items to be selected).

Example 5 Using an ordered selection

A business is selecting a chairperson and a deputy chairperson. The nominations are Adam, Caitlin, Jake, Ben and Lucy. The chairperson is selected first, then the deputy chairperson.a How many different selections are possible?b What is the probability of Lucy being elected as the

chairperson and Adam as the deputy chairperson?

Solution

1 Order is important in this event. Caitlin then Lucy is different from Lucy then Caitlin.

2 The first event is selecting the chairperson. There are 5 possible outcomes.

3 The second event is selecting the deputy chairperson. (There are 4 possible outcomes as one person has been selected.)


5 Number of favourable outcomes is 1. The total number of outcomes is 20.

6 Substitute into the probability formula.

a Ordered selection. Number of selections = 4

= 205 5 2 or P

b One selection out of 20 possible selections. P(LA)

LA)= =

n

n s

(

( )

1

20




Example 6 Using an ordered selection

The first, second and third speakers in a debate are selected randomly from a group of seven students.a How many different selections are possible?b Jack was in the group of seven students and selected as the first speaker. How many

different selections are possible for the second and third speakers?c Chloe was in the group of seven students and selected as the second speaker. How many

different selections are possible for the third speaker?d What is the probability of Jack, Chloe and Kerry being selected from the group of seven

students as the first, second and third speakers?

Solution

1 Order is important in this event.2 The first event is selecting the first speaker.

There are 7 possible outcomes.3 The second event is selecting the second

speaker. (6 possible outcomes as one person has been selected.)

4 The third event is selecting the third speaker. (5 possible outcomes as two people have been selected.)


6 Order is important in this event.7 The first event is selecting the second speaker.

There are 6 possible outcomes.8 The second event is selecting the third speaker.

(5 possible outcomes as one person has been selected.)


10 Order is important in this event.11 The first event (only event) is selecting the third

speaker. There are 5 possible outcomes.



a Ordered selection. Number of selections

= 6 = 2107 5 7 3or P

b Ordered selection. Number of selections

= 5 = 306 6 2or P

c Ordered selection. Number of selections

= 5 5 1or P

d One selection out of 210 possible selections.

n

n sP(JCK)

(JCK)

( )

1

210= =




Exercise 6C 1 Ava has eight balls, each labelled with a different number from 1 to 8. How many

arrangements are possible if selecting (without replacement):a 2 balls? b 3 balls? c 4 balls?d 5 balls? e 6 balls? f 7 balls?

2 Ethan has 26 cards, each labelled with a different letter from A to Z. How many arrangements are possible if selecting (without replacement):a 2 cards? b 3 cards? c 4 cards?

3 A school is selecting a captain and a vice captain. The nominations are Dylan, Ella, Holly, Patrick, Samuel and Tahlia. The captain is selected first, then the vice captain. How many different selections are possible?

4 The local community has nominated the best five gardens. There is a first prize and a second prize awarded to these gardens. How many different selections are possible?

5 There are 40 discs in a container each labelled with a number from 1 to 40. Two discs are removed and placed in order. How many different selections are possible?

6 How many arrangements of three letters can be made from the letters P, Q, R, S, T, U and V? The arrangement PQR is different from PRQ.

7 The letters of the word GRAFTON are written on cards and turned face down. Cards are selected at random.a How many two-letter arrangements can be made from this word?b How many three-letter arrangements can be made from this word?c How many four-letter arrangements can be made from this word?

8 There are 15 horses running in a race. Assume there are no dead heats.a How many different arrangements are possible for first and second place?b How many different arrangements are possible for first, second and third place?

9 A marathon has 12 runners.a How many different arrangements are possible for first and second place?b How many different arrangements are possible for first, second and third place?




Development

10 Joshuas class has 19 students. The principal needs to select four students from this class to represent the school at four different conferences. The students are selected at random with the first student chosen to attend conference A, the second student chosen to attend conference B, etc.a How many different selections are possible?b What is the probability that Joshua, Emily, Mia and Thomas will be selected to

attend the conferences A, B, C and D respectively?c There are two new enrolments to Joshuas class. How many selections are possible if

the new students are included in the calculation?d Joshua was selected to attend conference A. How many different selections are

possible for the other places, if the two new students are included?

11 A netball team is planning a raffle to raise money for a local charity. There are 100 raffle tickets, each labelled with a different number from 1 to 100. The raffle is awarding a first, second and third prize. Each ticket can win at most one prize.a How many different arrangements are

possible?b Unfortunately only 80 tickets were sold.

How many different arrangements are possible?

c The members of the netball team decided to include another prize. How many different arrangements are possible?

12 A three-digit number is formed from the digits 1, 2, 3, 4 and 5. No digit may be used more than once in the same number.a How many different numbers can be formed?b How many numbers are greater than 400?c How many numbers are less than 300?d What is the probability that the number formed is greater than 200?e What is the probability that the number formed is less than 500?




6.4 Unordered selectionsUnordered selections or a combination occurs when a selection is made from a group of items and the order is not important. For example, selecting three students from a class of 20 or choosing 6 numbers from 45 numbers in Lotto. The order of these selections is irrelevant. Consider the event of selecting two cards from four cards (1, 2, 3 and 4) without replacement. If the selection was a 2 followed by a 3 or a 3 followed by a 2 then the result is the same two cards. We are not interested in the order in which the objects were chosen, but merely the content that was selected.

Unordered selections

Selection is made from a group of items and the order is not important.

AB is the same as BA.

Combination nCr (n items available for selection and r items to be selected).

Example 7 Using an unordered selection

A teacher writes the vowels on the blackboard. A student randomly selects and erases two vowels from the blackboard.a How many possible selections are there?b What is the probability of selecting the O and

the U?

Solution

1 Order is not important in this event. Selecting an O then U is the same as a U then an O.

2 The first event is selecting the first letter. There are 5 possible outcomes.

3 The second event is selecting the second letter. (There are 4 possible outcomes as one letter has been selected.)

4 The number of combinations for 2 letters is 2 1 (OU and UO are the same).

5 Multiply the number of outcomes for each event and divide by the number of combinations.

6 The number of favourable outcomes is 1. The total number of outcomes is 10.


a Unordered selection. OU is the same as UO. Number of selections

=5 4

1= 10

25

2or C

b One selection out of 10 possible selections.

P(OU)OU)

= =

n

n s

(

( )

1

10




Example 8 Using an unordered selection

A pizza shop offers seven different toppings of pizza.a Riley chooses two toppings for a pizza. How

many different possible choices could he make?b Riley chooses three toppings for a pizza. How

many different possible choices could he make?c What is the probability of selecting ham,

pineapple and tomato?

Solution

1 Order is not important in this event. Ham and cheese is the same as cheese and ham.

2 The first event is selecting the first topping. There are 7 possible outcomes.

3 The second event is selecting the second topping. (There are 6 possible outcomes as one topping has been selected.)

4 The number of combinations for 2 toppings is 2 1.


6 The first event is selecting the first topping. There are 7 possible outcomes.

7 The second event is selecting the second topping. (There are 6 possible outcomes as one topping has been selected.)

8 The third event is selecting the third topping. (There are 5 possible outcomes as two toppings have been selected.)

9 The number of combinations for 3 toppings is 3 2 1 or 6.




a Unordered selection. Number of selections

C2or 7 2=

7 61

= 21

b Unordered selection. Number of selections

=7 6 5

1= 35

3 27

3or C

c One selection out of 35 possible selections.

n

n sP(HPT)

(HPT)

( )

1

35= =




Exercise 6D 1 What is the number of possible combinations from these selections?

a 2 items chosen from 7 items b 3 items chosen from 8 itemsc 4 items chosen from 6 items d 5 items chosen from 7 itemse 3 items chosen from 5 items f 6 items chosen from 11 items

2 Two teenagers are to be selected from a group of six teenagers to form a team. How many different teams can be formed?

3 An environmental committee is to be formed from a group of 15 students.a If the committee has two students, how many different selections are possible?b If the committee has four students, how many different selections are possible?c If the committee has six students, how many different selections are possible?

4 Bags are available in nine different colours.a Two bags are chosen at random.

How many different selections are possible?

b Three bags are chosen at random. How many different selections are possible?

c Four bags are chosen at random. How many different selections are possible?

d Five bags are chosen at random. How many different selections are possible?

e Six bags are chosen at random. How many different selections are possible?

5 How many different football teams of 11 players can be chosen from:a 12 players? b 13 players? c 14 players?d 15 players? e 16 players? f 17 players?

6 In the gambling game of Lotto you choose 6 numbers from 45 numbers.a How many different selections are possible?b What is the probability of winning Lotto with one game or selection?c A system 7 game involves selecting 7 numbers and receiving all six possible number

combinations. How many combinations of 6 are possible from 7 numbers?




Development

7 There are six cards, each labelled with a different number from 1 to 6. Two cards are to be drawn at random without replacement.a How many different selections are possible?b What is the probability of drawing a 1 and a 2 from the six cards?c A card with the number 7 is added to these six cards. How many selections are

possible if this new card is included in the calculation?d Three cards are drawn at random without replacement from the seven cards. How

many different selections are possible?

8 Five different flavours of ice-cream are available: chocolate, strawberry, vanilla, caramel and passionfruit.a How many different double-cone

selections are possible if two different flavours must be used?

b What is the probability of choosing chocolate and vanilla?

c How many different triple-cone selections are possible if three different flavours must be used?

d What is the probability that the chocolate scoop will be at the bottom of a triple cone?

e What is the probability of choosing chocolate, vanilla and strawberry?

f How many different triple-cone selections are possible if strawberry is unavailable?

9 A survey is to be conducted and three people are to be chosen from a group of twenty.a In how many different ways could the three be chosen?b If the group contains eight men and twelve women, how many groups containing

exactly one man are possible?

10 David chooses five cards from a normal pack of 52 playing cards.a How many different selections are possible?b How many five-card hands will have exactly one ace?c How many five-card hands will have exactly two aces?




6.5 Probability trees: Product ruleThe probability of two independent events occurring is equal to the product of the probability of each event. For example, when two unbiased coins are tossed the probability of throwing two heads is equal to the product of the probability of throwing a head with each coin, or 0.5 0.5 = 0.25. This result is shown by listing the sample space {HH, HT, TH or TT}. The probability of two heads is one out of four (0.25).

Probability of event A and B

P(AB) = P(A) P(B)

P(AB) Probability of event A and B (both events occurring)

P(A) Probability of event A

P(B) Probability of event B

To calculate the probability of two events occurring on a tree diagram, multiply the probabilities along each successive branch.

Example 9 Finding the probability of events A and B

A coin is tossed and a die thrown. A tree diagram is shown opposite.a What is the probability of obtaining a head and a 4?b What is the probability of obtaining a head and not a 4?

Solution

1 The first event is tossing a coin and the probability of a head is one out of two.

2 The second event is throwing a die and the probability of a 4 is one out of six.

3 Multiply the probability of both events.4 The first event is tossing a coin and the

probability of a head is one out of two.5 The second event is throwing a die and the

probability of not a 4 is five out of six.6 Multiply the probability of both events.

a P(Head and 4) =

=

1

21

12

16

b P(Head and not 4)1

25

12

= 56

=

Head4

Not 4

4

Not 4Tail

12

12

56

16

56

16




Example 10 Finding the probability of three events

Joshua has a 0.6 chance of winning a set of tennis against Harry. Find the probability of:

a Joshua winning three consecutive setsof tennis

b Harry winning three consecutive setsof tennis.

W0.6

0.6

0.4

0.6

0.4

0.6

0.4

0.4

L

W

L

W

L

W

L

0.6

0.4

W

L

0.6

0.4

W

L

0.6

0.4

W

L

1st 2nd 3rd

Solution

1 The first event is Joshua winning the first set (probability is 0.6).

2 The second event is Joshua winning the second set (probability is 0.6).

3 The third event is Joshua winning the third set (probability is 0.6).

4 Multiply the probability of the three events.5 The first event is Harry winning the

first set or Joshua losing the first set (probability is 0.4).

6 The second event is Harry winning the second set (probability is 0.4).

7 The third event is Harry winning the third set (probability is 0.4).

8 Multiply the probability of the three events.

a P(WWW) 0.60.216

= 0.6 0.6=

b P(LLL) 0.40.064

= 0.4 0.4=




Exercise 6E 1 What is the probability of the following outcomes?

a One tail when an unbiased coin is tossed once.b Two tails when an unbiased coin is tossed twice.c Two heads when an unbiased coin is tossed twice.

2 A box contains 7 black balls and 6 white balls. Two balls are drawn in succession from the box. The first ball is replaced before the second ball is drawn.a What is the probability of the first ball being black?b What is the probability of drawing two black balls?c What is the probability of drawing two white balls?

3 The probability that a set of traffic lights shows red, amber or green is equally likely. Aaron is travelling down a road that has two sets of traffic lights.a What is the probability that the

first set of traffic lights will be red?

b What is the probability that both sets of traffic lights will be red?

4 Daniel buys 7 tickets in a raffle in which 100 tickets were sold. Two different tickets are drawn for the first and second prizes.a What is the probability Daniel wins first prize?b What is the probability Daniel wins both prizes?c What is the probability Daniel does not win either prize?

5 Each time he shoots an arrow, the probability that Andrew hits the target is 4

9. He shoots

two arrows, one after the other.a What is the probability he hits the target with both arrows?b What is the probability he misses the target with both arrows?

HeadHead

Tail

Head

TailTail

12

12

12

12

12

12




6 In a particular group of students, the probability of a boy having blue eyes is 4

11 and

blond hair is 2

7. A boy is chosen at random from that group.

a What is the probability the boy is blue-eyed and has blond hair?b What is the probability the boy is blue-eyed but does not have blond hair?

7 There are 70 girls and 80 boys in Year 11 and 60 girls and 60 boys in Year 12. A student is chosen at random from each year.a What is the probability of choosing a boy from Year 11?b What is the probability of choosing a boy from Year 12?c What is the probability that the students in both years will be boys?

8 A deck of cards has 8 clubs and 6 spades. A second deck has 3 clubs and 7 spades. One card is selected at random from each deck. What is the probability of selecting two clubs?

9 One bag contains 6 cards numbered 1, 2, 3, . . . 6, and a second bag contains 10 cards lettered a, b, c, . . . , j. One card is drawn from each bag. Find the probability of drawing:a the number 5 and the letter d b an odd number and the letter cc an even number and a vowel d a number less than 5 and the

letter f .

10 In one bag there are 4 blue and 5 red balls and in a second bag there are 3 blue and 4 red balls. One ball is drawn from each bag.a What is the probability the balls drawn are both red?b What is the probability the balls drawn are both blue?c What is the probability of drawing a blue ball from the first bag and a red ball from

the second bag?d What is the probability of drawing a red ball from the first bag and a blue ball from

the second bag?

11 An unbiased coin and a die are tossed. Find the probability of obtaining a:a Tail on the coin and a 2 on the die.b Head on the coin and an even number on the die.c Tail on the coin and a number greater than 1 on the die.d Head on the coin and a number divisible by 3 on the die.




Development

12 Five men and seven women meet at a restaurant. However, some of the people need to leave.a One person is selected at random to leave the restaurant. What is the probability that

the person will be a female?b Two people are selected at random to leave the restaurant. What is the probability

that the two people selected are male?c Three people are selected at random to leave the restaurant. What is the probability

that the three people selected are female?

13 Mei owns three brown caps and four red caps. She selects one of the caps for herself at random and two other caps at random for friends.a What is the probability that Mei

selects a brown cap for herself?b Construct a tree diagram with

the correct probability on each branch.

c Calculate the probability the three caps are brown.

d Calculate the probability the three caps are red.

14 A jar contains 7 green and 5 yellow balls. Find the probability of drawing out balls alternating in colour starting with a green ball (without replacement) if there are:

a two draws b three draws c four draws.

15 Luke travels to Melbourne for the annual general meeting. He stays at one of three motels. Motel D is his favourite and he stays there on 60% of his visits to Melbourne. When he does not stay at Motel D, he is equally likely to stay at Motel E or F. Luke flips a coin on the first morning of every visit to decide whether he has a walk before breakfast. If the coin is heads he goes for a walk. If the coin is tails he stays in bed.a What is the probability of staying at Motel E?b List all the possible combinations of motel and whether he walks or stays in bed.c What is the probability Luke stays at Motel D and goes for a walk?d What is the probability Luke stays at Motel E and stays in bed?e What is the probability Luke stays at Motel F and goes for a walk?




6.6 Probability trees: Addition ruleThe probability of one event or a second event is equal to the sum of the probabilities of each event. For example, when two unbiased coins are tossed the probability of throwing two heads or two tails is equal to the sum of the probabilities of two heads and two tails or 0.25 + 0.25 = 0.50. This result is shown by listing the sample space {HH, HT, TH or TT}. The probability of two heads or two tails is two out of four (0.50).

Probability of event A or B

P(A or B) = P(A) + P(B)

P(A or B) Probability of event A or event B

P(A) Probability of event A

P(B) Probability of event B

To calculate the probability of one event or the second event on a tree diagram, add the probabilities for each event. The probability of each event is obtained by multiplying the probabilities along each successive branch.

Example 11 Finding the probability of event A or B

A container holds 4 pink and 3 green paper clips. Two clips are chosen at random. The first clip chosen is put back into the container before the second is chosen. A tree diagram is shown opposite.a What is the probability of both paper clips being

green?b What is the probability that the two paper clips are

pink and green?

Solution

1 Probability of a green paper clip is 3 out of 7 for both events.

2 Multiply the probability of both events.

3 Two outcomes are possible either pink and green or green and pink.

4 Calculate the probability of each outcome by multiplying the probabilities.

5 Add the probabilities of both events.6 Evaluate.

a Event A and B along one branch (GG)

P(GG)3

7

9

49=

37

=

b Event A or B two branches (PG + GP)

P(PG or GB) P(PG) P(GB)4

7

4

724

49

= +

= 37

+37

=

PinkPink

Green

Pink

GreenGreen

47

37

37

47

37

47




Example 12 Finding the probability of event A or B

Bailey has three aces and two kings face down on the table.a Draw a probability tree diagram for selecting two cards at

random.b What is the probability of Bailey selecting an ace with the

first card and a king with the second card?c What is the probability of Bailey selecting an ace with the

first or the second card, but not both?

Solution

1 Draw the first branches (first card). 2 Probability of an ace is 3 out of 5 cards

and the probability of a king is 2 out of 5 cards. Write these probabilities on the branches.

3 Draw the second branches (second card). 4 If an ace was selected as the first card,

there 2 aces and 2 kings remaining out of 4 cards. Write these probabilities on the branches.

5 If a king was selected as the first card, there are 3 aces and 1 king remaining out of 4 cards. Write these probabilities on the branches.

6 Probability of an ace is 3 out of 5 for the first card and 2 out of 4 for the second card.

7 Multiply the probability of both events. 8 Evaluate and simplify.

9 Two outcomes are possible: either ace and king or a king and an ace.

10 Calculate the probability of each outcome by multiplying the probabilities.

11 Add the probabilities of both events.12 Evaluate and simplify.

a

AceAce

King

Ace

KingKing

35

25

14

34

24

24

b Event A and B along one branch (AK)

P(AK)3

56

20

3

10

= 24

= =

c Event A or B two branches used (AK + KA)

P(AK or KA)2

4

3

412

203

5

=

35

+25

=

=




Exercise 6F 1 An unbiased coin is tossed twice. The outcomes are

shown in the tree diagram opposite.a What is the probability of two heads?b What is the probability of throwing a head on the

first throw and a tail on the second throw?c What is the probability of a head and a tail in any

order?d What is the probability of two heads or two tails?

2 A bag contains 9 balls. There are 5 blue balls and 4 red balls. Bhushan removes two balls from the bag one at a time. The first ball is placed back in the bag before the second ball is removed.a What is the probability of two blue balls?b What is the probability of first removing a blue ball

and then a red ball?c What is the probability of a blue ball and a red ball in

any order?d What is the probability of removing two blue balls or two red balls?

3 Melanoma, a type of skin cancer, will affect 7% of the male population.a What is the probability that a male selected at random will not suffer from

melanoma?b Two males are selected at random. What is the probability both males will suffer

from melanoma?c Two males are selected at random. What is the probability only one male will suffer

from skin cancer?

4 A, B and C are mutually exclusive events, and P(A) = 1

4, P(B) =

1

3 and P(C) =

2

7.

Find the value of the following probabilities.a P(A or B)b P(B or C)c P(A or C)

HeadHead

Tail

Head

TailTail

12

12

12

12

12

12

BlueBlue

Red

Blue

RedRed

59

49

49

59

49

59




5 Ryan participates in a dart competition. He throws two darts. The probability that he hits the bullseye on any one throw is 19%.a What is the probability that Ryan

misses the bullseye with his first throw?

b What is the probability that Ryan hits the bullseye once only?

c Calculate the probability that Ryan hits the bullseye with both throws.

6 Riley and Emma play two games of

squash. The probability of Riley winning one game of squash is 3

8.

a What is the probability of Emma winning both games of squash?b What is the probability of Riley winning both games of squash?c What is the probability of Riley winning at least one game of squash?

7 A discount shop has 13 DVDs for sale. Of these DVDs, 4 are rated G, 7 are rated PG and 2 are rated M.a Amy chooses two DVDs at random. The first DVD

is shown on a tree diagram. Complete the tree diagram for second DVD.

b What is the probability of choosing two DVDs that arerated G?

c Find the probability that Amy chooses two DVDs with thesame rating.

8 Jacob has 10 pens in a desk drawer. There are 6 black and 4 blue pens. Two pens are selected at random.a Construct a tree diagram. Label the probability of each outcome.b What is the probability of selecting a pair of pens that are different colours?c What is the probability of selecting a pair of pens that are the same colour?

9 A one deck of cards has 7 clubs and 5 spades. A second deck has 4 clubs and 9 spades. One card is selected at random from each deck. What is the probability of selecting two cards with the same suit?

10 A container has 6 white and 5 yellow tennis balls. Two balls are selected at random. What is the probability the two balls will be different colours?

G

PG

M

413

713

213




Development

11 There are 20 tickets sold in a raffle. Jordan has bought 3 tickets.a If there are two prizes, what is

the probability he wins at least one prize?

b If there are three prizes, what is the probability he wins at least one prize?

c If there are three prizes, what is the probability he wins exactly two prizes?

d If there are three prizes, what is the probability he wins three prizes?

12 A financial adviser predicts the sharemarket has a 0.6 chance of rising and a 0.4 chance of falling in any year.a Calculate the probability over the next two years that shares will:

i rise in both years ii rise in the first year and fall in the second yeariii rise in at least one of the two years.

b Calculate the probability over the next three years that shares will: i fall in all three years ii fall in at least two of the three yearsiii fall in at least one of the three years.

13 A pile of cards contains hearts and spades only. These occur in the ratio of 2 to 5. If three cards are chosen at random from the pile and each card is replaced before the next one is chosen, find the probability that:a exactly two are hearts b at least one is a spade.

14 The probability of a boy being born in a community is 40%. Find the probability that in a family of three children:a all the children are boys b all the children are girlsc there are two boys and a girl d there are two girls and a boy.

15 Three students are selected at random from Year 12.a What is the probability all the students were born in the month of July?b What is the probability two of the students were born in the month of July?c What is the probability one of the students was born in the month of July?




6.7 Expected outcomesThe expected outcome is the number of times the outcome should occur. It may not equal the

actual results. For example, when a coin is tossed the probability of getting a head is 1

2.

Hence, if a coin is tossed 100 times the expected number of heads is 50 or 1

2 100. Clearly,

if a coin is tossed 100 times it may not result in exactly 50 heads. However, the larger the number of trials the closer the expected outcome will be to the actual results.

Expected outcomes

Expected outcome is the number of times the outcome should occur.

Expected outcome = P(E) Number of trials

P(E) Probability of the event.

The expected outcome may not be a whole number. It is an estimate of what to expect. For example,

when a die is tossed the probability of getting a six is 1

6. Hence, if a die is tossed 100 times the

expected number of sixes is 1

6 100 or 16

2

3. Clearly, it is not possible to have

2

3 of an outcome.

However, the expectation is that the number of outcomes will be a whole number close to 162

3.

Example 13 Finding the expected outcome

Two coins are tossed 120 times and the results recorded.a What is the expected number of outcomes for two heads?b What is the expected number of outcomes for a head and a tail?

Solution

1 Calculate the probability of two heads. 2 Number of favourable outcomes is 1 (HH).

The total number of outcomes is 4 (HH, HT, TH, TT).

3 Write the formula for expected outcomes. 4 Substitute into the formula. 5 Evaluate.

6 Calculate the probability for a head and a tail. 7 Number of favourable outcomes is 2 (HT, TH).

The total number of outcomes is 4 (HH, HT, TH, TT).

8 Write the formula for expected outcomes. 9 Substitute into the formula.10 Evaluate.

a

P(HH)1

4=

Expected outcomesEP( ) Number of trials

1

4

=

= 120

= 30

b P(HT or TH)2

4

1

2= =

Expected outcomes

EP( ) Number of trials1

2

=

= 120

= 60

6.1




Exercise 6G 1 The probability of a red traffic light at an intersection is

1

3. How many red traffic lights

are expected on a trip that passes through 54 intersections?

2 The probability of a person living in a certain community of developing melanoma is four out of nine. There are 1404 people living in this community. What is the expected number of people who will develop melanoma?

3 Andrew and Caitlin are planning to have five children. A genetic counsellor has calculated they have a 40% chance of having a child with green eyes. How many of Andrew and Caitlins children are expected to have green eyes?

4 Jessica is a goal shooter for her netball team. The probability that she scores a goal is 88%. This year she had 225 attempts at goal.a How many goals would you

expect Jessica to score this year?b How many goals would you

expect Jessica to miss this year?

5 Jake is a professional golfer who has a 78% chance of breaking par. He plays 150 golf courses in a year. How many times would you expect Jake to break par in a year?

6 The probability of a worker in an industrial plant having an accident is 0.12. The industrial plant employs 175 workers. What is the expected number of accidents?

7 A die is tossed 480 times and the results recorded.a What is the probability of throwing a 4?b How many 4s are expected?c What is the probability of throwing an odd number?d How many odd numbers are expected?e What is the probability of throwing a number greater than 2?f How many numbers greater than 2 are expected?g What is the probability of throwing a number divisible by 3?h How many numbers divisible by 3 are expected?




Development

8 Five cards (ace, king, queen, jack and 10) are placed face down on a table. One card is selected at random and replaced. A second card is then selected at random. This experiment is repeated 200 times.a What is the probability of selecting two

aces?b How many double aces are expected?c What is the probability of selecting an ace

followed by a king?d How many aces then kings are expected?e What is the probability of exactly one of

the cards being a 10?f How many single 10s are expected?

9 A three-digit number is selected from cards labelled 3, 4 and 5. The first card selected is the hundreds digit, the second card is the tens digit and the third card is the units digit. The cards are selected without replacement. This selection is repeated 30 times.a What is the probability the number starts with the digit 3?b How many numbers starting with the digit 3 are expected?c What is the probability the number is 453?d How many 453s are expected?e What is the probability the number ends with a 4 or a 5?f How many numbers ending with a 4 or 5 are expected?

10 A bag contains 6 yellow discs and 5 red discs. Two discs are drawn in succession from the bag. The first disc is not replaced before the second disc is drawn. This process is repeated 352 times.a How many of the first discs are expected to be yellow discs?b How many of the first discs are expected to be red discs?c How many double yellow discs are expected?d How many double red discs are expected?e How many are expected to have a first disc yellow and a second disc red?f How many are expected to have a first disc red and a second disc yellow?




11 Create the spreadsheet below.

a Cell C8 has a formula that multiplies cells B8 and B4. Enter this formula.b Fill down the contents of C8 to C11 using the formula in cell C8.c Change the number of trials from 230 to 800. Observe the change in C8:C11.

12 There are 240 families with three children.a How many of these families are expected to have three boys?b How many of these families are expected to have exactly one boy?c How many of these families are expected to have exactly two boys?d How many of these families are expected to have no boys?

13 Two cards are selected at random from a normal playing pack with replacement. This experiment is repeated 2704 times with the cards being replaced each time.a What is the expected number of times the result would be two spades?b What is the expected number of times the result would be two aces?c What is the expected number of times the result would be two picture cards?d What is the expected number of times the result would be two cards with a number

less than 9?

14 Two dice are tossed simultaneously onto a table. This event is repeated 144 times.a On how many occasions would you expect the result to be a 6 then a 1?b On how many occasions would you expect the result to be two 3s?c On how many occasions would you expect the result to be two odd numbers?




6.8 Expected valueExpected value indicates the expected outcome to be achieved in an event. It is calculated by multiplying each outcome by its probability and then adding all these results together. Financial expectation is the expected value when the event involves money. The financial outcome is positive if money will be won and negative if money will be lost. For example, if a coin is tossed and the financial outcome of a head is winning $2 and a tail is losing $1 (-$1),

then the financial expectation is 1

2 $2 +

1

2 (-$1) or $0.50. It is expected over an extended

period of time to win $0.50 per game.

Expected value and financial expectation

Expected value = Sum all results [P(E) outcome]

Financial expectation = Sum all results [P(E) Financial outcome]

P(E) Probability of the event or financial outcome

Financial outcome Positive if winning and negative if losing

Positive financial expectation profit Negative financial expectation loss

Example 14 Finding the expected value

What is the expected number of car thefts in one day if there is a 30% chance of no car thefts, a 50% chance of one car theft and a 20% chance of two car thefts?

Solution

1 Write the formula for expected value.

2 Substitute into the formula.3 Evaluate.

Expected value= Sum[P( ) outcome]

=30 50

E

100 100 0 + 1

+ 2 =

20

1000 9.

Example 15 Finding the financial expectation

Find the financial expectation of a ticket in a raffle. The raffle has 300 tickets and there is one prize worth $250.

Solution

1 Write the formula for financial expectation.

2 Substitute into the formula.3 Evaluate.

Financial expectation= Sum[P( ) Financial outcome]

=1

E

300 250

+ 0 =

299

3000 83$ .




Example 16 Finding the financial expectation

Roulette is a game played in a casino. The winning number in roulette is determined by the position of a small ball spun on a roulette wheel. A player can bet on any of the numbers from 1 to 36 or a combination of these numbers, such as 1 to 18 or the even numbers. This is shown on the roulette table opposite. When a zero is rolled the casino wins.

What is the financial expectation for the following games? Answer correct to the nearest cent.a Zachary bets $20 on red. When a red number

is spun, he wins $20 otherwise he loses $20.b Amy bets $20 on 112. When a number from

1 to 12 is spun she wins $40, otherwise she loses $20.

c What conclusions can be made about the above game?

Solution

1 Write the formula for financial expectation.

2 Substitute into the formula. The financial outcome is positive for a win (20) and negative for a loss (-20).

3 Evaluate.4 Write the formula for financial

expectation.5 Substitute into the formula. The

financial outcome is positive for a win (40) and negative for a loss (-20).

6 Evaluate.7 Negative financial expectation

indicates a loss.

a Financial expectation

=

= 20 +

Sum[P( ) Financial outcome]

18

E

37

119

37

0 54

20

=

$ .

b Financial expectation

=

= 40

Sum[P( ) Financial outcome]

12

E

37

++ 20

=

25

37

0 54

$ .

c Both Zachary and Amy are expecting to lose $0.54 per game over a large number of bets.




Exercise 6H 1 A landscape gardener mows 20 lawns per day on sunny days and 15 lawns per day on

cloudy days. The weather is sunny 65% and cloudy 35% of the time. How many lawns can he expect to mow per day?

2 A car dealer gets daily complaints about his cars. The probabilities of receiving 0, 1, 2, 3 or 4 complaints are 0.1, 0.3, 0.4, 0.1 and 0.1 respectively. What is the expected number of complaints per day?

3 An unbiased coin is tossed. What are the financial expectations of these games?a A tail wins $10 and a head loses $10.b A tail wins $10 and a head loses $5.c A tail wins $5 and a head loses $10.

4 A fair die is thrown. What are the financial expectations of these games?a A 6 wins $50 and not a 6 loses $25.b An even number wins $20 and an odd number loses $15.c A 1 or a 2 wins $30 and a number greater than 2 loses $20.

5 A business has a 30% chance of making $200 000, a 20% chance of making $100 000, a 15% chance of making $50 000, a 25% chance of breaking even and a 10% chance of losing $200 000. Calculate the financial expectation of the small business.

6 Mia plays a game in which she has a 20% chance of winning $40, a 50% chance of winning $1 and a 30% chance of losing $3. What is Mias financial expectation when playing this game?

7 Jackson has a 70% chance of selling his house for $300 000 and a 30% chance of selling the house for $320 000. What is the expected sale value of Jacksons house?

8 A lottery has a $600 000 first prize and a $240 000 second prize. There were 360 000 tickets sold in the lottery. What is the financial expectation, to the nearest cent, of each ticket?

9 Sarah plays a game by throwing two unbiased dice. The rules of the game are: Sarah wins $25 if there are two 4s. Sarah wins $2.50 if there is only one 4. Sarah loses $10 if there are no 4s.

a What is the probability of throwing a double 4?b What is the financial expectation of this game? Answer to the nearest cent.




Development

10 In basketball, you can earn 3 points for a shot and 1 point for a free throw. If Jacks probability of getting a 3-point shot is 40% and 80% for a free throw, what is his expected score for the game?

11 Hayley is playing a game at an amusement park. There is a 0.1 probability that she will score 10 points, a 0.2 probability that she will score 20 points and a 0.7 probability that she will score 30 points. How many points can Hayley expect to receive by playing the game?

12 Daniel plays a game by selecting one of four aces. The rules of the game are as follows. Daniel wins $40 by selecting a red card. Daniel has no result by selecting a club. Daniel loses $100 by selecting a spade.He plays the game 160 times and replaces the card after each game.a How many times would he expect to win $40?b How many times would he expect to lose $100?c What is the financial expectation of this game? Answer to the nearest cent.

13 Paige plays a game by throwing three coins. The rules of the game are as follows. Paige wins $60 if there are three tails. Paige wins $15 if there are two tails. Paige loses $30 if there is one or no tails.

She plays the game 120 times and replaces the card after each game.a How many times would she expect to win $60?b How many times would she expect to lose $30?c What is the financial expectation of this game? Answer to the nearest cent.

14 Four hundred raffle tickets are sold at $3 each. The first prize is $500, second prize is $250 and there are ten third prizes each consisting of a gift card.a Blake buys two tickets in the raffle. What is the probability that he wins first prize?b Ignore the gift card and determine the financial expectation of the raffle.c What is the value of the gift card for the raffle to be fair?

15 Emily is designing a game with four possible results. She has decided on three of these results. What must be the value of the loss in Result D in order for the financial expectation of this game to be $0?

Probability Financial outcome

Result A 20% Win $100

Result B 30% Win $60

Result C 40% Win $30

Result D



Review239Chapter 6 Multistage events and applications of probability

Chapter summary Multistage events and applications of probability Study guide 6

Multistage events Outcomes for each event are listed downthe page with the events extending across the page. Each event is a new branch of the tree. The sample space is listed on the right-hand side.

Probability

En E

n S

Probability (Event)Number of favourable outcomes

Total number of outcomes

P( )( )

( )

=

=

Number of arrangements Number of outcomes (two events) = p qp Number of outcomes of the first eventq Number of outcomes of the second event

Ordered selections Selection from a group of items and the order is important. AB is different from BA.Permutation nP

r

n items available for selection and r items to be selected

Unordered selections Selection from a group of items and the order is not important. AB is the same as BA.Combination nC

r

n items available for selection and r items to be selected

Probability of independent

events A and B

P(AB) = P(A) P(B) P(AB) Probability of event A and event B (both) P(A) Probability of event A P(B) Probability of event B

Probability of event A or B P(A or B) = P(A) + P(B) P(A or B) Probability of event A or event B P(A) Probability of event A P(B) Probability of event B

Expected outcomes The number of times the outcome should occur

Expected outcome = P(E) Number of trials

Expected value Expected value = Sum all results [P(E) outcome]

Financial expectation Financial exp = Sum all results [P(E) Financial outcome] Financial outcome Positive (win) and negative (loss)

H

T

H

H

T

T

HH

HT

TH

TT

1st 2nd



Revi

ewHSC Mathematics General 2240

1 Two dice are rolled. What is the probability that only one die shows a 6?

A 1

36 B

1

6 C

5

18 D

11

36

2 Daniel is deciding on the number of digits required for an invoice number. How many more numbers are available if Daniel uses five digits (e.g. 78012) compared to using four digits (e.g. 1275)?A 10 B 90 000 C 10 000 D 100 000

3 In how many different ways can the letters of the word MATHS be arranged in a row?A 6 B 21 C 24 D 120

4 There are 20 runners in a marathon. How many different selections are possible for first and second place? Assume there are no dead-heats.A 39 B 190 C 380 D 400

5 A business has nominated the best five employees. There is a first prize and a second prize awarded to these employees. How many different selections are possible?A 10 B 20 C 25 D 120

6 Jasmine has two packets of jelly babies. Each packet contains two orange and four red jelly babies. Jasmine takes one jelly baby from each packet without looking. What is the probability both jelly babies are orange?

A 1

15 B

1

9 C

1

6 D

1

2

7 A cupboard contains 7 white mugs and 4 black mugs. A mug is taken at random from the cupboard, and then returned to the cupboard after its colour has been noted. A second mug is then taken at random from the cupboard. What is the probability both mugs are the same colour?

A 28

121 B

49

121 C

56

121 D

65

121

8 What is the probability of throwing at least one 6 if a die is thrown twice?

A 1

36 B

1

18 C

11

36 D

25

36

9 Two unbiased coins are tossed 100 times. Which calculation illustrates the expected number of times you would get a tail and a head?

A 1

4100 B 1

2100 C

1

3100 D 1

4 200

Sample HSC Objective-response questions



Review241Chapter 6 Multistage events and applications of probability

Sample HSC Short-answer questions

1 A two-digit number is formed using the digits 2, 4, 6 and 8. The same number cannot be used twice. The first digit chosen is the tens digit and the second digit chosen is the units digit.a Find the sample space from the tree diagram.b What is the probability that the number starts with a 4?c What is the probability that the number ends with an 8?d What is the probability that the number formed is 62?

2 Four cards (heart, diamond, spade and club) are placed face down on a table. One card is selected at random and the result recorded. This card is returned to the table. A second card is then selected at random. What is the total number of outcomes?

3 In how many ways can Alyssa, Bridget, Chun, David and Eddie stand in a queue?

4 Two people are selected at random to represent the local community from Blake, Chris, Debbie, Emily, Fran and Grace. Order of the selection is important.a How many different ways of selection are

possible?b What is the probability of selecting Chris then

Fran?c The community has decided to send four

people instead of two people. How many different selections are now possible?

5 A, B and C are independent events, and P(A) = 1

3, P(B) =

1

2 and P(C) =

2

5 . Find the value of the following probabilities.

a P(AB) b P(BC) c P(ABC)

6 There are 14 participants in a competition.a How many different arrangements are possible for first and second place?b How many different arrangements are possible for first, second and third place?c How many different arrangements are possible for first, second, third and fourth

place?

7 How many different committees of seven can be chosen from:a 10 people? b 11 people? c 12 people?



Revi

ewHSC Mathematics General 2242

8 A bag contains ten cards. The cards are marked with the letters A to J. Three cards are drawn at random without replacement and used in that order to form a word.a How many possible selections are there?b What is the probability of selecting D as the first letter?c What is the probability of drawing DIG?d What is the probability of drawing a D with the first letter or with the second letter?

9 A bag contains four cards labelled P, Q, R and S. A card is chosen and removed from the bag at random. A second card is then chosen and removed from the bag.a What is the probability of selecting a P with the first card?b What is the probability of selecting PQ?c What is the probability of selecting a P with the first or second card?

10 A coin is tossed and a die is thrown.a What is the probability of tossing a head and throwing an even number?b What is the probability of tossing a tail and throwing a 1 or 5?

11 Luke has a 75% chance of getting his first serve into play during a tennis match. If he has two serves, find the probability of getting:a no serves into play b at least one serve into play.

12 A box contains 4 blue and 9 green discs. Two discs are chosen from the box without replacement. Find the probability of selecting:a 2 blue discs b 2 green discsc 2 discs the same colour d 2 discs of a different colour.

13 Scott has four queens and three jacks face down on the table. He draws two cards at random without replacement. What is the probability of selecting a queen with either the first or the second card but not both?

14 The probability of a couple having a baby with red hair is 331

3%. If they have six children,

how many children with red hair are expected?

15 A probability of a dog having heartworm is 3

8. If there are 896 dogs in the local

community, how many of them would you expect to have heartworm?

16 Joshua plays a game with two dice. He gains $30 if he gets two odd numbers and $5 if he gets an odd and an even number, but he loses $35 if he ends up with two even numbers. What is the financial expectation of this game?

Challenge questions 6



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