Page 69 - Probability Demystified

58 CHAPTER 4 The Multiplication Rules

ANSWERS

1. Independent. Tossing a coin has no effect on drawing a card.
2. Dependent. In most cases, driving on ice will increase the probability
of having an accident.
3. Dependent. Since the first ball is not replaced before the second ball is
selected, it will change the probability of a specific second ball being
selected.
4. Independent. To the best of the author’s knowledge, no studies have
been done showing any relationship between hat size and I.Q.
5. Independent. The outcome of the first coin does not influence the
outcome of the second coin.

Multiplication Rule I

Before explaining the first multiplication rule, consider the example of tossing
two coins. The sample space is HH, HT, TH, TT. From classical probability
1
theory, it can be determined that the probability of getting two heads is ,
4
since there is only one way to get two heads and there are four outcomes in
the sample space. However, there is another way to determine the probability
of getting two heads. In this case, the probability of getting a head on the first
1
1
toss is , and the probability of getting a head on the second toss is also .
2 2
So the probability of getting two heads can be determined by multiplying
1 ¼ This example illustrates the first multiplication rule.
1
1
2 2 4
Multiplication Rule I: For two independent events A and B, PðA and BÞ¼
PðAÞ PðBÞ.
In other words, when two independent events occur in sequence, the prob-
ability that both events will occur can be found by multiplying the probabil-
ities of each individual event.
The word and is the key word and means that both events occur in
sequence and to multiply.

EXAMPLE: A coin is tossed and a die is rolled. Find the probability of
getting a tail on the coin and a 5 on the die.

SOLUTION:
1
1
1
1
Since PðtailÞ¼ 1 and Pð5Þ¼ ; Pðtail and 5Þ¼ PðtailÞ Pð5Þ¼ ¼ .
2 6 2 6 12
Note that the events are independent.