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)
P(1 boy and 1 girl)
 ways to   ways to 

  

 choose boys   choose girls 
 ways to 


choose
two


 ways to   ways to 

  

 choose boys   choose girls 
 ways to 


choose
two


2
C0 7 C2 1  21 21


36
36
9 C2
2
C1 7 C1 2  7 14


36 36
9 C2
P(2 boys and 0 girls)
 ways to   ways to 

  

 choose boys   choose girls 
 ways to 


choose
two


2
Notice that if we add up all of the probabilities we get 1.
21 14 1 21  14  1 36
 


1
36 36 36
36
36
C2 7 C0 1 1 1


36 36
9 C2
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)
In how many different ways
can you pick the six winning
numbers?
The probability of winning first
prize is then given by the
following:
49 C6 
49! 49  48  47  46  45  44  43!

 13,983,816
6!43!
(6  5  4  3  2 1)  43!
6 C6 
6! 6!
 1
6!0! 6!
C6
1

13,983,816
49 C6
6
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)
In how many different ways
can you pick 5 of the 6
winning numbers?
The probability of winning
second prize is then given by
the following:
49 C6 
49! 49  48  47  46  45  44  43!

 13,983,816
6!43!
(6  5  4  3  2 1)  43!
 ways to pick  ways to pick 
6! 43!


 6 C5 43 C1 

 6  43  258
5
winners
1
loser
5
!

1
!
1
!

42
!



6
C5 43 C1
258
43
1



13,983,816 2,330,636 54,200
49 C6
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 to pick  ways to pick 
6! 43!


 6 C4 43 C2 

 15  903  13,545
4!2! 2!41!
 4 winners  2 losers 
6
C4 43 C2
13,545
645
1



13,983,816 665,896 1,032
49 C6