Page 73 - Probability Demystified

62 CHAPTER 4 The Multiplication Rules

1
probability of getting a specific card is 52 , but the probability of getting a
specific card on the second draw is 1 since 51 cards remain.
51

EXAMPLE: Two cards are selected from a deck and the first card is not
replaced. Find the probability of getting two kings.

SOLUTION:
4
The probability of getting a king on the first draw is 52 and the probability
of getting a king on the second draw is 3 , since there are 3 kings left and
51
51 cards left. Hence the probability of getting 2 kings when the first card is
not replaced is 4 3 ¼ 12 ¼ 1 .
52 51 2652 221

When the two events A and B are dependent, the probability that the
second event B occurs after the first event A has already occurred is written as
P(B | A). This does not mean that B is divided by A; rather, it means and is
read as ‘‘the probability that event B occurs given that event A has already
occurred.’’ P(B | A) also means the conditional probability that event B occurs
given event A has occurred. The second multiplication rule follows.

Multiplication Rule II: When two events are dependent, the probability of
both events occurring is PðA and BÞ¼ PðAÞ PðB j AÞ

EXAMPLE: A box contains 24 toasters, 3 of which are defective. If
two toasters are selected and tested, find the probability that both are
defective.

SOLUTION:
Since there are 3 defective toasters out of 24, the probability that the first
1
toaster is defective is 3 ¼ . Since the second toaster is selected from the
24 8
remaining 23 and there are two defective toasters left, the probability that
it is defective is 2 . Hence, the probability that both toasters are defective is
23
1 1
3 2 1
PðD and D Þ¼ PðD Þ PðD jD Þ¼ 4 ¼
1
2
1
1
2
24 8 23 92