Thinking Mathematically

Probability with the Fundamental Counting Principle, Permutations,

and Combinations

Example Probability and Permutations

Five groups in a tour, Offspring, Pink Floyd, Sublime, the Rolling Stones, and the Beatles, agree to determine the order of performance based on a random selection. Each band’s name is written on one of five cards. The cards are placed in a hat and then five cards are drawn out, one at a time. The order in which the cards are drawn determines the order in which the bands perform. What is the probability of the Rolling Stones performing fourth and the Beatles last?

Solution

We begin by applying the definition of probability to this situation.

P(Rolling Stones fourth, Beatles last) =(permutations with Rolling Stones fourth, Beatles last)

(total number of possible permutations)We can use the Fundamental Counting Principle to find

the total number of possible permutations.

5 4 3 2 1 = 120

Solution cont.We can also use the Fundamental Counting Principle to

find the number of permutations with the Rolling Stones performing fourth and the Beatles performing last. You can choose any one of the three groups as the opening act. This leaves two choices for the second group to perform, and only one choice for the third group to perform. Then we have one choice for fourth and last.

3 2 1 1 1 = 6There are six lineups with Rolling Stones fourth and

Beatles last.

Solution cont.

Now we can return to our probability fraction.

P(Rolling Stones fourth, Beatles last) =(permutations with Rolling Stones fourth, Beatles last)

(total number of possible permutations)

= 6/120 = 1/20The probability of the Rolling Stones

performing fourth and the Beatles last is 1/20.

Example Probability and Combinations: Winning the Lottery

Florida’s lottery game, LOTTO, is set up so that each player chooses six numbers from 1 to 53. If the six numbers chosen match the six numbers drawn randomly twice weekly, the player wins (or shares) the top cash prize. With one LOTTO ticket, what is the probability of winning this prize?

Solution

Because the order of the six numbers does not matter, this is a situation involving combinations. We begin with the formula for probability.

P(winning) = number of ways of winning total number of possible

combinationsWe can use the combinations formula to find

the total number of combinations.

Solution cont.

There are nearly 23 million number combinations possible in LOTTO, and only one is a winning combination.

P(winning) = number of ways of winning total number of possible

combinations

= 1/22,957,480

53C6 =53!

(53−6)!6!=

53!47!6!

=53 • 52 • 51• 50 • 49 • 48 • 47!

47!• 6 • 5• 4 • 3 • 2 • 1=22,957,480

Example Probability and Combinations

A club consists of five men and seven women. Three members are selected at random to attend a conference. Find the probability that the selected group consists of:

a. three men.

b. one man and two women.

Solution

We begin with the probability of selecting three men.

P( 3 men)=number of ways of selecting 3 men

total number of possible combinations

12C3 = 12!/((12-3)!3!) = 220

5C3 = 5!/((5-3)!(3!)) = 10

P(3 men) = 10/220 = 1/22

Solution part b

We set up the fraction for the probability that the selected group consists of one man and two women. P(1 man and 2 women) = number of ways of selecting 1 man and 2 women

total number of possible combinations

We know the denominator is 12C3 = 220.Next we move to the numerator of the

probability fraction.

Solution part b cont.

The number of ways of selecting r = 1 man from n = 5 men is

5C1 = 5!/(((5-1)!1!) = 5

The number of ways of selecting r = 2 women from n=7 women is

7C2 = 7!/((7-2)!2!) = 21

Solution part b cont.

By the Fundamental Counting Principle, the number of ways of selecting 1 man and 2 women is

5C1 7C2 = 5 21 = 105Now we can fill in the numbers in our probability

fraction. P(1 man and 2 women) = number of ways of selecting 1 man and 2 women

total number of possible combinations

= 105/220 = 21/44

Thinking Mathematically

Probability with the Fundamental Counting Principle, Permutations,

and Combinations