Page 118 - Probability Demystified

CHAPTER 6 The Counting Rules 107

EXAMPLE: An identification card consists of 3 digits selected from 10 digits.
Find the probability that a randomly selected card contains the digits 1, 2,
and 3. Repetitions are not permitted.

SOLUTION:
3!
The number of permutations of 1, 2, and 3 is P ¼ ð3 3Þ! ¼ 3! ¼ 3 2 1 ¼ 6
3
3
1
0!
The number of permutations of 3 digits each that can be made from 10
digits is
10! 10! 10 9 8 7!
P ¼ ¼ ¼ ¼ 720
10 3
ð10 3Þ! 7! 7!
Hence the probability that the card contains 1, 2, and 3 in any order is
6 1
¼ 0:008
720 120

PRACTICE

1. In a classroom, there are 10 men and 6 women. If 3 students are
selected at random to give a presentation, find the probability that
all 3 are women.
2. A carton contains 12 toasters, 3 of which are defective. If four toast-
ers are sold at random, find the probability that exactly one will be
defective.
3. If 100 tickets are sold for two prizes, and one person buys two tickets,
find the probability that that person wins both prizes.
4. A committee of 3 people is formed from 6 nurses and 4 doctors. Find
the probability that the committee contains 2 nurses and one doctor.
The committee members are selected at random.
5. If 5 cards are dealt, find the probability of getting 4 of a kind.

ANSWERS

1. There are 6 C 3 ways to select 3 women from 6 women.
6! 6! 6 5 4 3!
C ¼ ¼ ¼ ¼ 20
6 3
ð6 3Þ!3! 3!3! 3! 3 2 1