Chapter 3

Section 3.4

Combinatorics and Probability

Counting Methods The counting methods of combination, permutation and Fundamental Counting Principle can be used to find probabilities.

A class consists of 2 boys and 7 girls. If two students are picked at random find the probabilities of each given below.

P

(0 boys and 2 girls)  

ways choose to boys

    

ways choose to girls

   

ways choose to two

 

P

(1 boy and 1 girl)  

ways choose to boys

    

ways choose to girls

   

ways choose to two

 

P

(2 boys and 0 girls)  

ways choose to boys

    

ways choose to girls

   

ways choose to two

  2

C

0  7

C

2 9

C

2  1  21  36 21 36 2

C

1  7 9

C

2

C

1  2  7 36  14 36 2

C

2 9  7

C

2

C

0  1  1 36  1 36 Notice that if we add up all of the probabilities we get 1.

21  14 36 36  1 36  21  14  1  36 36 36  1

Lotteries The structure of a lottery is given by two numbers. Ohio's Lottery (Classic Lotto) is described as a 6/49 lottery. That is from a set of 49 balls numbered 1-49 there are 6 that are randomly chosen. You get to pick the six numbers you think will match the 6 that are randomly chosen.

Winning first prize means that you pick all 6 of the numbers that were chosen.

What is the probability of winning first prize?

In how many different ways can 6 lottery balls be chosen from 49? ( Ohio Lottery ) 49

C

6  49 !

6 !

 43 !

 49  48  47  46  45  44  43 !

 13 , 983 , 816 ( 6  5  4  3  2  1 )  43 !

In how many different ways can you pick the six winning numbers?

6

C

6  6 !

6 !

 0 !

 6 !

6 !

 1 The probability of winning first prize is then given by the following: 6

C

6 49

C

6  1 13 , 983 , 816

In Ohio's 6/49 lottery what is the probability of matching exactly 5 of the 6 numbers correctly?

In how many different ways can 6 lottery balls be chosen from 49? ( Ohio Lottery ) 49

C

6  49 !

6 !

 43 !

 49  48  47  46  45  44  43 !

 13 , 983 , 816 ( 6  5  4  3  2  1 )  43 !

In how many different ways can you pick 5 of the 6 winning numbers?

 

ways

5

to pick winners

   

ways

1

to pick loser

   6

C

5  43

C

1  6 !

5 !

 1 !

 43 !

1 !

 42 !

 6  43  258 The probability of winning second prize is then given by the following: 6

C

5  43

C

1 49

C

6  258 13 , 983 , 816  43 2 , 330 , 636  1 54 , 200 What is the probability of matching exactly 4 of the 6 numbers correctly?

In how many different ways can you pick 4 of the 6 winning numbers?

The probability of winning third prize is then given by the following:  

ways

4

to pick winners

   

ways

2

to pick losers

   6

C

4  43

C

2  6 !

4 !

 2 !

 43 !

2 !

 41 !

 15  903  13 , 545 6

C

4 49  43

C

6

C

2  13 , 545 13 , 983 , 816  645 665 , 896  1 1 , 032