(13 – 1) The Counting Principle and Permutations

Learning targets: To use the fundamental counting principle to count the number of ways an event can happen

To use permutations to count the number of ways an event can happen

The Fundamental Counting Principle

(also known as the multiplication rule for counting) 

If a task can be performed in n1 ways, and for each of these a second task can be performed in n2 ways, and for each of the latter a third task can be

performed in n3 ways, ..., and for each of the latter a kth task can be

performed in nk ways, then the entire sequence of k tasks can be

performed in  n1 • n2 • n3 • ... • nk  ways.

I do: (ex) A sandwich stand has four kinds of meat, ham, turkey, roast beef, and chicken. Also they have three kinds of bread, white, wheat, and rye. How many choices do you have?

Ham

Turkey

Roast beef

Chicken

White

Wheet

Rye

White

Wheet

Rye

White

Wheet

Rye

White

Wheet

Rye

3 choices with ham

3 choices with turkey

3 choices with roast beef

3 choices with chickenTotal 12 choices

4 3 12

You do: (ex) You have 5 T-shirts and 3 pair of pants. How many choices of outfit do you have?

We do: (ex) You can choose 3 different flowers out of 7 with the same price. How many choices you have?

The first choice out of 7: 7 choices

The second choice out of 6: 6 choices

The third choice out of 5: 5 choices

Total choices: (7)(6)(5)=210 choices

(13 – 1) continued Using Permutations

Definition:

A permutation an order of n objects. (Order is the matter)

It is the same as the Fundamental Counting Principle.

r

!P

( )!n

n

n r

Notation:

The number of permutations of r objects taken from a group of n distinct objects is given by:

Note: A symbol ! is called “factorial”.

n! = n (n – 1)(n – 2)(n – 3)…(1)

(ex) 5! = 5 (4) (3) (2) (1) = 120

Remember: 0! = 1

I do (ex) What is the total number of possible 4-letter arrangements of the letters m, a, t, h,  if each letter is used only once in each arrangement? 

1st letter 4 choices

2nd letter 3 choices

3rd letter 2 choices

4th letter 1 choice

Total 4! = 24 arrangements

We do (ex) Number plate

The number plate of a car consists of 3 letter and 4 numbers. How many ways can they arrange without repetition?

1st letter: 26 choices

2nd letter: 25 choices

3rd letter: 24 choices

AND

1st number 10 choices

2nd number 9 choices

3rd number 8 choices

4th number 7 choices

26 3

26!P

(26 3)!

10 4

10!P

(10 4)!

Total arrangements:

You do: (ex) Joleen is on a shopping spree.  She buys six tops, three shorts and 4 pairs of sandals.  How many different outfits consisting of a top, shorts and sandals can she create from her new purchases?

A pizza shop runs a special where you can buy a large pizza with one cheese, one vegetable, and meat for $9.00. You have a choice of 7 cheeses, 11 vegetables, and t meats. Additionally, you have a choice of 3 crusts and 2 sauces. How many different variations of the pizza special are possible?

Permutation with repetition (distinguishable permutation):

The number of distinguishable permutations of n objects where one object is repeated q1 times, another is repeated q2 times, and so on is:

I do: (ex) Find the number of distinguishable permutations of the letter in Cincinnati?

Cincinnati has 10 letters. (n)

i is repeated 3 times (q1)

n is repeated 3 times (q2)

c is repeated 2 times (q3)

1 2

!

( !)( !)...( !)k

n

q q q

=10!

(3!)(3!)(2!)

1 2

!

( !)( !)...( !)k

n

q q q

We do: (ex) Find the number of distinguishable permutations of the letter in Mississippi?

Mississippi has 11 letters. (n)

i is repeated 4 times (q1)

s is repeated 4 times (q2)

p is repeated 2 times (q3)

You do: (ex) Find the number of distinguishable permutations of the letter in Calculus?

Find the number of distinguishable permutations of the letters in the word.

a) PUPPY b) LETTER c) MISSOURI

d) CONNECTICUT

Combinations

Definition:

A combination is a selection of r objects from a group of n objects where the order is not important.

Notation: and

!

( )! !n r

nc

n r r

n rc

I do: (ex) There are 12 boys and 14 girls in Mrs. Brown’s math class.  Find the number of ways Mrs. Brown can select a team of  3 students from the class to work on a group project.  The team is to consist of 1 girl and 2 boys.     

It is a combination question because order, or position, is not important.

To select 2 boys from 12 is: n = 12 and r = 2

AND

To select 1 girl from 14 is: n = 14 and r = 1

Total ways:

!

( )! !n r

nc

n r r

!

( )! !n r

nc

n r r

We do: (ex) You can select at most 3 items out of 10 items on your pizza. How many different types of pizza do you have as your choices?

10 0n rc c•At most means: 0, 1, 2, OR 3 items The choice of 0 items:

The choice of 1 item:

The choice of 2 items:

The choice of 3 items

Total choices:You add if the situation is OR.

You do: (ex) In how many ways can 7 cards be dealt from a deck of 52 cards if you want 4 red and 3 black cards.

(Hint: how many red cards and black cards are there? What is n then?)

<Try this> In how many ways can you obtain all 7 red or all 7 black?

(12 – 2) continued Using Binomial Theorem

Complete the next three row of Pascal's Triangle.

Where

n r

nc

r

I do: (ex) Expand

4( )x y

We do: (ex) Expand

6( 2)x You do: (ex) Expand

4( 2 )x y

(13 – 3) Probability

Definition:

The probability of an event is a number between 0 and 1 that shows the chance will happen.

Theoretical probability (probability): When an event A will happen out of all outcomes happen equally, likely.

Theoretical probability is:

number of outcoms in A( )

total number of outcomsP A

I do: (ex) What is the probability to get an odd number when you roll a die?

A die has 6 faces (1~6)

An odd number are 1, 3, or 51

(rolling 1)6

P

1(rolling 3)

6P

1(rolling 5)

6P

(rolling an odd number)P

We do (ex) You shuffle the numbers 1~6 and place them. What is the probability to get 6 as the 1st number?

Total number to fill 6 places: 6!

1st place 2nd place 3rd 4th 5th 6th

6

1(5!)(gettubg 6 as the 1st place)

6!P

We do: (ex) Teresa and Julia are among 10 students who have applied for a trip to Washington, D.C. Two students from the group will be selected at random for the trip. What is the probability that Teresa and Julia will be the 2 students selected?

2 2

10 2

c

c

You do (ex) There are 20 cards (from 1 through 20). What is the probability to get a perfect square is chosen?

<Try this> (1) What is the probability to get a face card out of 52-card deck if you randomly pick one card.

(2) A math teacher is randomly distributing 15 rulers with centimeter labels and 10 rulers without centimeter labels. What is the probability that the first ruler she hands out will have centimeter labels and the second ruler will not have labels?

<try this> (3) One bag contains 2 green marbles and 4 white marbles, and a second bag contains 3 green marbles and 1 white marble. If Trent randomly draws one marble from each bag, what is the probability that they are both green?

Complement of probability

Definition:

The complement of event is that not in the event

Notation:A’ is called “complement of event A”

Probability of complement of A is: P(A’) = 1 – P(A)

I do: (ex) Two dice are tossed. What is the probability that the sum is not 8?

P (sum is not 8) = 1 – P (sum is 8)

Sum is 8: (2 + 6), (3 + 5), (4 + 4), (5 + 3), (6 + 2)

total 5 ways

Each die has 6 faces and there are two, so

total # of event when two dice are tossed: 62

Then and

5(sum is not 8) =1

36P

5(sum is 8) =

36P

We do: (ex) On a certain day the chance of rain is 80% in San Francisco and 30%in Sydney. Assume that the chance of rain in the two cities is independent. What is the probability that it will not rain in either city?

A 7% B 14% C 24% D 50%(0.2)(0.7) = 0.14

We do: (ex) The probabilities that Jamie will try out for various sports and team positions are shown in the chart below.

Jamie will definitely try out for either basketball or baseball, but not both. The probability that Jamie will try out for baseball and try out for catcher is 42%. What is the probability that Jamie will try out for basketball? A 40% B 60% C 80% D 90%

<try this> Two six-sided dice are rolled. Find the probability of the given event.

a) The sum is greater than or equal to 5

b) The sum is less than or equal to 10.

c) The sum is greater than 2.

Probability of Independent & Dependent Events

Definition:

An independent event is an event that is not affected by other event.

•If A and B are independent events, then the probability that both A and B occur is: P(A and B) = P(A)P(B).

Probability of Independent events

I do (ex) You toss a die twice. What a probability to get 1 on the first toss and 3 on the second toss.

1P (to get 1) =

61

P (to get 3) = 61 1 1

P (to get 1 and 3) = 6 6 36

We do (ex) A basket has 5 red, 4 yellow, and 3 green. You are picking 3 balls with replacement. If you pick Red, Yellow, and Green in order, you will be the winner. What is the probability to be a winner?

1st pick 2nd pick 3rd pick

5( )

12P red

4( )

12P yellow 3

( )12

P green

Therefore, 5 4 3

( )12 12 12

P RYG

You do (ex) You randomly select two cards from a 52-card deck. What is the probability that the first card is not a face and second card is a face card with replacement.

<Try this> One bag contains 2 green marbles and 4 white marbles, and a second bag contains 3 green marbles and 1 white marble. If you randomly draw one marble from each bag, what is the probability that they are both green?

Probability of Dependent events:

Definition:

A dependent event is an event in which one occurrence affects the other occurrence. The probability that B will occur under that A has occurred is called “Conditional probability” and noted by:

P(BA) read as “probability of B given A”

Formula:

If A and B are dependent events, then the probability that both A and B occur is:

P(A and B) = P(A) P(BA)

I do (ex) (from CST sample question 2010)

A box contains 7 large red marbles, 5 large yellow marbles, 3 small red marbles, and 5 small yellow marbles. If a marble is drawn at random, what is the probability that it is yellow, given that it is one of the large marbles?

P( yellow| large) = # of large of yellow

Total # of large

We do (ex) You and two friends are at a restaurant for lunch. There are 8 dishes with the same price. What is the probability that each of you orders a different dish?

Let three dishes are A, B, and C

1st person’s 2nd person’s 3rd person’s8

( )8

P A 7( )

8P B

6( )

8P C

8 7 6( , ,& )

8 8 8P A B C

You do (ex) A math teacher is randomly distributing 15 plastic rulers and 10 wooden rulers. What is the probability that the first ruler she hands out will be a wooden one, and the second ruler will be a plastic one?

1st pick 2nd pick

10 ( )

25P wooden

15 ( )

24P plastic

10 15 ( & )

25 24P wooden plastic

(13 – 3 continued) Probability Involving AND and OR

I do (ex)

Find the probability to draw a diamond or a face card from a deck of cards.

13 12 3

52 52 52

13 diamond cards : P(D)

12 face cards: P(F)

3 diamond and face cards : P(D & F)

P(D or F) = P(D) + P(F) – P(D & F) =

We do: One card is drawn from a deck of cards. What is the probability that the card is a black or an ace?

P(B)

P(A)

P(B & A)

P(B or A)

You do (ex)

One card is drawn from a deck of cards. What is the probability that the card is either a face or heart?

P(F)

P(H)

P(F & H)

P(F or H)

1.Mean ( ):

2.Median:

3.Standard deviation: ia a measure of how each value in a data set varies from the mean. And the notation is

sum of the data values

number of data valuesMean

middle value or mean of the two middle valueMedian x

x

(sigma)

2the sume of (eadch data - mean value)=

n

2

1

( )=

n

n

ii

X X

or Where xi is each data as 1, , nX X

Statistics:

Definition: Statistics are numerical values used to summarize and compare sets of data. Measures of central tendency are:

I do (ex) Find the mean and the standard deviation for the values:

48.0, 53.2, 52.3, 46.6, 49.9

Mean 48.0 53.2 52.3 46.6 49.950.0

5X

Standard deviation:

2

1

2 2 2 2 2

( )=

n

(50 48.0) (50 53.2) (50 52.3) (50 46.6) (50 49.9)

5

31.12.5

5

n

ii

X X

We do (ex) Keith found the mean and standard deviation of the set of the numbers given below. If he adds 5 to each number, what is the standard deviation?

3, 6, 2, 1, 7, 5

For the data given:

244

6X

28

6

After 5 is added:54

96

X

2 2 2 2 2 2(9 8) (9 11) (9 7) (9 6) (9 12) (9 10)

6

28

6

<try this> Find the mean and the standard deviation for each set of values.

(a)5, 6, 7, 3, 4, 5, 6, 7, 8 (b) 13, 15, 17, 18, 12, 21, 10