If and are two events of a sample space of a random experiment, where the number of the event occurring is 16, the number of the event occurring is 12, the number of both events and occurring is 6, and , then find .

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Example

If and are two events of a sample space of a random experiment, where the number of the event occurring is 16, the number of the event occurring is 12, the number of both events and occurring is 6, and , then find .

Solution

The probability of event occurring is equal to the number of
favorable outcomes divided by the total number of possible
outcomes in the sample space, Find the probability of event
in terms of the total number of possible outcomes in the
sample space as follows:

The probability of event occurring is equal to the number of
favorable outcomes divided by the total number of possible
outcomes in the sample space, Now find the probability of
event in terms of the total number of possible outcomes in the
sample space.

The probability of both event and event occurring is equal
to the number of outcomes favorable for both events divided by the
total number of possible outcomes in the sample space, Next,
find the probability of both event and event occurring in
terms of the total number of possible outcomes in the sample
space.

The notation refers to the probability of either
event or event occurring, or both. If and are two
non-mutually exclusive events, then where the notation refers to the
probability of events and occurring together. Now use this
fact to determine the total number of possible outcomes in the
sample space.

Since the total number of possible outcomes in the sample space is
44, it follows that ,
and Since and are complementary, the sum of their probabilities is
equal to 1. Now use this fact to find the probability of

De Morgan's Second Law states that where the notation
means the non-occurrence of or the non-occurrence of , or
both. Since it follows that