Transcript Chapter 3
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