Page 76 - Probability Demystified

CHAPTER 4 The Multiplication Rules 65

known that some outcomes in the sample space have occurred or that some
outcomes cannot occur. When conditions are imposed or known on events,
there is a possibility that the probability of the certain event occurring may
change. For example, suppose you want to determine the probability that a
house will be destroyed by a hurricane. If you used all houses in the United
States as the sample space, the probability would be very small. However, if
you used only the houses in the states that border the Atlantic Ocean as the
sample space, the probability would be much higher. Consider the following
examples.

EXAMPLE: A die is rolled; find the probability of getting a 4 if it is known
that an even number occurred when the die was rolled.

SOLUTION:

If it is known that an even number has occurred, the sample space is reduced
1
to 2, 4, or 6. Hence the probability of getting a 4 is since there is one chance
3
in three of getting a 4 if it is known that the result was an even number.

EXAMPLE: Two dice are rolled. Find the probability of getting a sum of 3 if
it is known that the sum of the spots on the dice was less than six.

SOLUTION:
There are 2 ways to get a sum of 3. They are (1, 2) and (2, 1), and there are
10 ways to get a sum less than six. They are (1, 1), (1, 2), (2, 1), (3, 1), (2, 2),
(1, 3), (1, 4), (2, 3), (3, 2), and (4, 1); hence, P(sum of 3|sum less than 6) ¼
2 ¼ .
1
10 5
The two previous examples of conditional probability were solved using
classical probability and reduced sample spaces; however, they can be solved
by using the following formula for conditional probability.
The conditional probability of two events A and B is
PðA and BÞ
PðAjBÞ¼ :
PðBÞ
P(A and B) means the probability of the outcomes that events A and B
have in common. The two previous examples will now be solved using the
formula for conditional probability.

EXAMPLE: A die is rolled; find the probability of getting a 4, if it is known
that an even number occurred when the die was rolled.