Page 117 - Probability Demystified

106 CHAPTER 6 The Counting Rules

Since there are 13 cards in a suit, there are 13 C 5 ways to get a flush in one
suit, and there are 4 suits, so the number of ways to get a flush is

13! 13!
4 C ¼ 4 ¼ 4
13
5
ð13 5Þ!5! 8!5!
13 12 11 10 9 8!
¼ 4
8! 5 4 3 2 1
¼ 5148
There are 52 C 5 ways to select 5 cards.
52! 52! 52 51 50 49 48 47!
52 C ¼ ¼ ¼ ¼ 2,598,960
5
ð52 5Þ!5! 47!5! 47! 5 4 3 2 1
5148
PðflushÞ¼ ¼ 0:00198 or about 0:002; which is about one
2,598,960
chance in 500:

EXAMPLE: A student has a choice of selecting three elective courses for the
next semester. He can choose from six humanities or four psychology
courses. Find the probability that all three courses selected will be humanities
courses assuming he selects them at random.

SOLUTION:
Since there are six humanities courses, and the student needs to select three of
them, there are 6 C 3 ways of doing this:
6! 6! 6 5 4 3!
C ¼ ¼ ¼ ¼ 20
6 3
ð6 3Þ!3! 3!3! 3! 3 2 1
The total number of ways of selecting 3 courses from 10 courses is 10 C 3 .
10! 10! 10 9 8 7!
10 C ¼ ¼ ¼ ¼ 120
3
ð10 3Þ!3! 7!3! 7! 3 2 1
Hence, the probability of selecting all humanities courses is
20 1
¼ 0:167
120 6

There is one chance in 6 that he will select all humanities courses if he
chooses them at random.