MODULE 9 WORKBOOKProbability

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7 May 2019

Contents

p 3

p 14

p 24

p 29

Lesson 1

Lesson 2

Apply (Non-calculator)

Apply (Calculator)

2

Module 9

Probability PODS Before continuing, make sure you have watched and have access to the following Pods.To find these, simply log in to GCSEPod and enter the codes given into the search bar (or click on the titles below).

1. Measuring Probability | MATHS-26-001

2. Calculating Probability | MATHS-26-002

3. Equally Likely Outcomes | MATHS-26-004

4. Relative Frequency | MATHS-26-003

5. Calculating Probability | MATHS-27-001

6. Venn Diagrams | MATHS-14-001

7. Tree Diagrams | MATHS-27-002

Lesson 1

Single event probability

PODS

1. Measuring Probability | MATHS-26-001

2. Calculating Probability | MATHS-26-002

3. Equally Likely Outcomes | MATHS-26-004

4. Relative Frequency | MATHS-26-003

3

Quiz

Watch the pod and answer the following questions.

1. The probability of an event happening is indicated with an arrow on the probabilityscale below. What is the probability of this event happening?

a) 0.2

b) 0.4

c) 0.1

d) 2

………………………………………………………………………………………………………………………………………………………………..

2. The probability of an event happening is indicated with an arrow on the probabilityscale below. What is the probability of this event NOT happening?

a) 0.2

b) 0.1

c) 0.8

d) 0.9

………………………………………………………………………………………………………………………………………………………………..

4

3. The probabilities of 4 events happening are indicated with arrows on the probabilityscale below:

Which of the probabilities indicates an event that has an even chance of happening or not happening?

a) P(A)

b) P(B)

c) P(C)

d) P(D)

………………………………………………………………………………………………………………………………………………………………..

4. The probabilities of 4 events happening are indicated with arrows on the probabilityscale below:

Which event has a probability of 4/5?

a) A

b) B

c) C

d) D

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5

5. Which cross indicates the probability of throwing a 2 on a fair 6 sided dice?

a) A

b) B

c) C

d) D

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6. Which of the following is not the probability of not choosing a red counter from thebag?

a) 2/10

b) 1/5

c) 0.2

d) 1/4

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6

7. The events A, B and C are mutually exclusive and exhaustive.The probabilities of events A and B are P(A) = 0.3 and P(B) = 0.2.

Calculate P(C).

a) 0.5

b) 0.06

c) 0.6

d) 0.1

………………………………………………………………………………………………………………………………………………………………..

8. The events A, B and C are mutually exclusive. The probabilities of these eventshappening are: P(A) = 0.15, P(B) = 0.3 and P(C) = 0.55.

Calculate P(A or B).

a) 0.045

b) 0.25

c) 0.55

d) 0.45

………………………………………………………………………………………………………………………………………………………………..

9. The overall number of goals scored by a school team last season are given in thetable below:

Which of the following is NOT the relative frequency of them scoring 2 goals?

a) 4/25

b) 0.16

c) 40%

d) 16%

………………………………………………………………………………………………………………………………………………………………..

7

10. Which letter indicates the probability of the event shown on the spinner?

a) A

b) B

c) C

d) D

………………………………………………………………………………………………………………………………………………………………..

8

Practise

Practise single event probability.

Do not use a calculator.

1. A counter is picked at random out of the following box.What is the probability that it is blue or green?

………………………………………………………………………………………………………………………………………………………………..

2. Jodie walks to school on Monday, Tuesday and Thursday. She gets the bus the other 2days. On any given day, what is the probability that she does not get the bus?

………………………………………………………………………………………………………………………………………………………………..

3. Patrick scores a goal 5 times when he makes 20 attempts. How many would youexpect him to score if he makes 60 attempts?

………………………………………………………………………………………………………………………………………………………………..

4. The probability that it is going to rain tomorrow is 0.64. What is the probability that it isnot going to rain?

………………………………………………………………………………………………………………………………………………………………..

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5. There are red, blue, purple and green beads in a bag. All beads have an equally likelychance of being chosen.

P(red) = 0.32

P(blue) = 0.2

P(purple) = 0.15

What is the probability of choosing a green?

………………………………………………………………………………………………………………………………………………………………..

10

Practise

Practise single event probability.

You may use a calculator.

1. A counter is picked at random from the collection shown below. It is replaced andagain, a counter is picked. This is repeated until a counter has been picked 50 times.How many times would you expect a blue to have been picked?

………………………………………………………………………………………………………………………………………………………………..

2. A spinner is spun a number of times; the number of times the spinner lands on eachsection is recorded in the table below:

What is the relative frequency of landing on blue?

………………………………………………………………………………………………………………………………………………………………..

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3. A spinner is spun a number of times; the number of times the spinner lands on eachsection is recorded in the table below:

If you spin this spinner 300 times, how many times would you expect it to land on a red?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

4. The number of goals per match scored by a school team overall last season are givenin the table below:

The team play 32 games next season. In how many games would you expect them to score 0 goals?

………………………………………………………………………………………………………………………………………………………………..

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5. Sam plays cricket. He can either win, lose or draw. The probability that he wins is 3/7.The probability of losing is 24%. What is the probability that he draws? Give youranswer correct to 2 decimal places.

………………………………………………………………………………………………………………………………………………………………..

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Lesson 2

Combined events

PODS

5. Calculating Probability | MATHS-27-001

6. Venn Diagrams | MATHS-14-001

7. Tree Diagrams | MATHS-27-002

Quiz

Watch the pod and answer the following questions.

1. 2 fair dice are thrown. Which of the following is not the probability of both diceshowing the same number?

a) 6/36

b) 1/6

c) 0.16

d) 6/30

………………………………………………………………………………………………………………………………………………………………..

2. A coin is flipped 2 times and the result is recorded. Which of the sample spacescorrectly shows all the possible outcomes?

a) {HH, HT, TH, TT}

b) {HT, TH}

c) {HH, HT, TT}

d) {TH, HT, TH, HT}

………………………………………………………………………………………………………………………………………………………………..

˙

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3. A coin is flipped 2 times and the result is recorded. What is the probability of flipping 2heads?

a) 0.5

b) 0.25

c) 0.75

d) 1/3

………………………………………………………………………………………………………………………………………………………………..

4. Cards numbered 1 to 20 are put into a hat. If 1 card is drawn at random, what is theprobability that it is a multiple of 3?

a) 3/10

b) 6/14

c) 3/7

d) 3/20

………………………………………………………………………………………………………………………………………………………………..

5. The Venn diagram below shows the number of students participating in cricket andtennis lunchtime clubs. Calculate P(C).

a) 8/19

b) 16 / 29

c) 10 / 16

d) 16 / 39

………………………………………………………………………………………………………………………………………………………………..

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15

6. The Venn diagram below shows the number of students participating in cricket and

tennis lunchtime clubs. Calculate P(C ᴜ T).

a) 14 / 36

b) 7/18

c) 17/18

d) 20 / 36

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

7. The Venn diagram below shows the number of people who watch comedy and

romance films in an office. Calculate P(C ∩ R).

a) 31/39

b) 16/39

c) 15/40

d) 5/13

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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8. Which of the following events are not independent?a) Taking a toffee from a bag of mixed sweets then, having eaten the first toffee,

taking another toffee.b) Picking a queen from a pack of cards then, having replaced it, picking another

queen.c) Flipping a coin and getting a tail, and then flipping it again and getting a head.d) Rolling a 1 on a dice and then rolling again and getting another 1.

………………………………………………………………………………………………………………………………………………………………..

9. The chance that Evan is late for school on any given day is ¼. What is the probabilitythat he will be late on the first day and the second day of school?

a) 1/16

b) 1/4

c) 1/2

d) 1/8

………………………………………………………………………………………………………………………………………………………………..

10. What is the chance of rolling a total of 8 when 2 dice are rolled?

a) 1/6

b) 1/12

c) 1/12

d) 5/36

………………………………………………………………………………………………………………………………………………………………..

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Practise

Practise combined events probability.

Do not use a calculator.

1. A dice is thrown, and a coin is flipped. Write out the sample space for this and use it tocalculate the probability of getting a Head and an Even Number.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

2. Two dice are thrown and the sum of their scores is recorded.Complete the sample space diagram below:

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3. There are 30 people in a class. 22 of them buy food from the café at lunch time. 18bring a packed lunch. All of them either bring packed lunch or buy food from the caféor do both.a) Represent this information in the Venn diagram below:

b) What is the probability that a student picked at random brings a packed lunch andbuys food?

………………………………………………………………………………………………………………………………………………………………..

4. There are 3 blue and 4 red counters in a bag. Joe takes one out, records its colour,replaces it and picks another.a) Complete the tree diagram:

b) What is the probability that he chooses 2 blue counters?

………………………………………………………………………………………………………………………………………………………………..

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5. On her way to school, Joni passes 2 crossings. The probability that she can walkstraight over the crossing is ¼. The probability that the crossing says ‘don’t walk’ is ¾.Show this information in a tree diagram and use it to calculate the probability of onlyone crossing saying ‘don’t walk’.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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Practise

Practise combined events probability.

You may use a calculator.

1. 3 boys and 4 girls have opted to go on a school trip, however, there are only 2 placesavailable. The teacher will choose 1 girl and 1 boy to attend the trip.

The list of students is: Bill, Catherine, Andrew, Dora, Elizabeth, Gail and Fred.

a) List all of the combinations of students that can be picked.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

b) What is the probability that both Catherine and Fred are chosen?

………………………………………………………………………………………………………………………………………………………………..

2. The table shows how children from year 7 travel to school:

Calculate the probability that a student chosen at random does not travel to school by walking. Give your answer as a decimal correct to 2 decimal places.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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3. 72 students attended a summer sports club where they could choose to focus onfootball, cricket or tennis.

Out of the 21 children who chose football, 9 were boys. 15 out of the 40 boys chosetennis and 6 girls chose cricket.

a) Complete the 2-way-table for this information:

b) Calculate the probability of a student chosen at random being a girl who isfocusing on tennis.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

4. Sienna carries out a survey in her school to find out what type of music 400 peoplelike listening to. 178 like listening to pop music. 298 like listening to rock music. 108 likelistening to both pop and rock music. Some people did not like either kind of music.a) Display this information in a Venn diagram:

b) Use your diagram to calculate the probability of a student picked at random likingboth rock and pop music.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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5. There are 45 red counters and 28 blue counters in a bag. A counter is taken out atrandom, its colour recorded and then replaced. A second counter is then taken fromthe bag.a) Complete the tree diagram below for this information:

b) Use your diagram to calculate the probability of getting the same colouredsweets.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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Apply

Apply what you have revised about probability.

Do not use a calculator.

Remember

Probability is out of 1 – all of the possible outcomes will add up to 1.

You can give your answer as a decimal or a fraction – check if the question specifies this.

Always draw a tree diagram for combined events.

Multiply across your branches and write these values at the end of your branch.

Add up the probabilities for a combination of outcomes using the values at the end of your branches.

1. There are some red, blue and purple counters in a box. 1 counter is chosen at random.The probability of picking a red counter is 0.45. The probability of choosing a blue

counter is 0.35. The probability of choosing a purple counter is "#$

.

Calculate the value of x.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

2. There are 20 cards placed in a box, each numbered from 1-20. A card is chosen atrandom. Calculate:

a) P(6)

………………………………………………………………………………………………………………………………………………………………..

b) P(even)

………………………………………………………………………………………………………………………………………………………………..

c) P(Square number)

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d) P(greater than 15)

………………………………………………………………………………………………………………………………………………………………..

e) P(less than 10 and greater than 15)

………………………………………………………………………………………………………………………………………………………………..

3. A fruit machine has 3 reels. Each reel shows a circle, a triangle or a square. When spun,each shape has an equal chance of being shown. The player wins a prize if all 3shapes match. Complete the sample space below, showing all of the outcomes anduse it to calculate the probability of winning a prize.

TTT, TTS, TTC, TST, TSS, TSC, TCT, TCS, TCC, STT, STS, STC, SST, SSS, SSC, SCT, SCS, SCC…

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

4. There are red, blue, purple and green beads in a bag. All beads have an equally likelychance of being chosen.

P(red) = 0.3

P(blue) = 0.2

P(purple) = 0.15

If there are 100 beads in the bag, how many are green?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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5. There are 40 sweets in a bag. 12 are chocolate, 11 are mint and the rest are toffee. Julestakes one sweet from the bag.

Calculate the probability that:

a) Jules takes a toffee

………………………………………………………………………………………………………………………………………………………………..

b) Jules takes a chocolate or a toffee

………………………………………………………………………………………………………………………………………………………………..

c) Jules does not take a chocolate

………………………………………………………………………………………………………………………………………………………………..

6. Matt has a box containing 10 balls that are coloured red, blue and yellow. Theprobability that he chooses a red ball is 1/2 and the probability that he chooses ayellow ball is 2/5. He selects a ball at random.a) What is the probability of Matt choosing a blue ball?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

b) Matt adds more yellow balls so that the chance of choosing a red is now 1/3. Howmany yellow balls has he added?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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7. Zane spins a spinner and records his results in a table:

Given that there are 7 equal sections on Zane’s spinner, how many of each colour do you think there are? You must give reasons for your answer.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

8. 2 fair dice are thrown and the product of their scores is recorded.a) Complete the sample space diagram:

b) What is the probability of the score being a square number?

………………………………………………………………………………………………………………………………………………………………..

c) What is the probability of the score being greater than 15?

………………………………………………………………………………………………………………………………………………………………..

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9. Owen likes to play card games. The chance that he wins when playing poker is ½. Thechance of him losing a game of rummy is 1/3. He plays poker then rummy and theresults of the games are independent of each other.a) Display this information in a tree diagram.

b) Calculate the probability of him losing at least one of his games.

………………………………………………………………………………………………………………………………………………………………..

10. Sophie has a bag of 10 sweets containing 7 toffees. The rest are chocolate. She takesone out, eats it, and then takes another.a) Complete the tree diagram showing this information:

b) Calculate the probability of choosing 2 toffees.

………………………………………………………………………………………………………………………………………………………………..

c) Calculate the probability of both sweets being different.

………………………………………………………………………………………………………………………………………………………………..

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Apply

Apply what you have revised about probability.

You may use a calculator.

Remember

Probability is out of 1 – all of the possible outcomes will add up to 1.

You can give your answer as a decimal or a fraction – check if the question specifies this.

Always draw a tree diagram for combined events and check your values carefully.

Multiply across your branches and write these values at the end of your branch.

Add up the probabilities for a combination of outcomes using the values at the end of your branches.

1. There are red, blue, purple and green beads in a bag. All beads have an equally likelychance of being chosen.

P(red) = 0.12

P(blue) = 0.24

P(purple) = 0.16

If there are 50 beads in the bag, how many are green?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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2. Year 9 students are choosing their options for GCSE. They must choose one of eitherPE, Art or Music. 102 people (64 of whom were boys) chose PE. Of the 127 girls in year9, 52 chose music. 52 of the 250 year 9 students chose Art.a) Display this information in a two way table:

b) A student is picked at random from all of the students in Year 9. What is theprobability that they are a girl who takes PE?

………………………………………………………………………………………………………………………………………………………………..

c) Calculate the probability that the student chosen takes music.

………………………………………………………………………………………………………………………………………………………………..

d) Calculate the probability that the student chosen is a girl.

………………………………………………………………………………………………………………………………………………………………..

3. There are 200 students in year 11. 107 take French and 113 take German. 12 students donot study a language.a) Display this information in a Venn diagram:

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b) Calculate P(French ∩ German)

………………………………………………………………………………………………………………………………………………………………..

c) Calculate P(Not French)

………………………………………………………………………………………………………………………………………………………………..

d) Calculate P(French ᴜ German)

………………………………………………………………………………………………………………………………………………………………..

4. The two spinners below are spun and their scores are multiplied together.If you score more than 10 you win a prize.

a) Complete the sample space diagram to show this information:

b) Clara says “there is an even chance of winning and losing in this game”.Is Clara correct? You must explain your answer.

………………………………………………………………………………………………………………………………………………………………..

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………………………………………………………………………………………………………………………………………………………………..

5. There are 75 counters in a bag. They are pink, purple or white. There are twice asmany purple as pink, and 10 more white than purple. If a counter is taken at randomfrom the bag, what is the probability that it is pink or purple?

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

6. James plays rugby. If it is raining, the chance that he wins his game is 0.7. If it is notraining, the chance that he loses is 0.6. The chance of it raining on the day of his nextmatch is 0.3.a) Complete the tree diagram below:

b) Calculate the probability of James winning his rugby match.

………………………………………………………………………………………………………………………………………………………………..

7. There are red, blue and green beads in a bag. The ratio of red to blue beads is 1:3 andthe ratio blue to green is 5:7. Calculate the probability of taking a blue bead from thebag at when a bead is taken from the bag at random.

………………………………………………………………………………………………………………………………………………………………..

………………………………………………………………………………………………………………………………………………………………..

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8. A council is investigating what kind of housing it should build to keep up with demandin its local area. The pie chart shows the type of house that are in the town.

a) What fraction of the housing in the town is flats?

………………………………………………………………………………………………………………………………………………………………..

b) The survey shows that 36,000 of the existing homes in the town are semi-detached houses. The council has identified that the town needs to have 150,000homes altogether. If it aims to maintain the existing proportions of each type, howmany terraced houses should be built?

………………………………………………………………………………………………………………………………………………………………..

9. Ben likes to play card games. The chance that he wins when playing poker is 0.6. Thechance of him losing a game of rummy is 0.7. He plays one game of each and theresults of the games are independent of each other.a) Display this information in a tree diagram.

b) Calculate the probability of him winning at least one of his games.

………………………………………………………………………………………………………………………………………………………………..

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10. There are 14 girls and 17 boys in a class. 2 students are going to be chosen at randomto represent the class on a school trip.a) Complete the tree diagram with this information:

b) What is the probability that the two students chosen will be of the same gender?

………………………………………………………………………………………………………………………………………………………………..

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