Page 123 - Probability Demystified

112 CHAPTER 6 The Counting Rules

14. At a used book sale, there are 6 novels and 4 biographies. If a person
selects 4 books at random, what is the probability that the person
selects two novels and two biographies?
a. 0.383
b. 0.562
c. 0.137
d. 0.429
15. To win a lottery, a person must select 4 numbers in any order from
20 numbers. Repetitions are not allowed. What is the probability that
the person wins?
a. 0.0002
b. 0.0034
c. 0.0018
d. 0.0015

Probability Sidelight

THE CLASSICAL BIRTHDAY PROBLEM
What do you think the chances are that in a classroom of 23 students, two
students would have the same birthday (day and month)? Most people would
think the probability is very low since there are 365 days in a year; however,
the probability is slightly greater than 50%! Furthermore, as the number of
students increases, the probability increases very rapidly. For example, if
there are 30 students in the room, there is a 70% chance that two students
will have the same birthday, and when there are 50 students in the room, the
probability jumps to 97%!
The problem can be solved by using permutations and the probability
rules. It must be assumed that all birthdays are equally likely. This is not
necessarily true, but it has little effect on the solution. The way to solve the
problem is to find the probability that no two people have the same birthday
and subtract it from one. Recall PðEÞ¼ 1 PðEÞ.
For example, suppose that there were only three people in the room.
Then the probability that each would have a different birthday would be

365 364 363 365 P 3
¼ ¼ 0:992
365 365 365 ð365Þ 3

The reasoning here is that the first person could be born on any day of the
year. Now if the second person would have a different birthday, there are