The Complement Rule In Probability Theory: Calculation And Examples

complement rule

P(A^c) = 1 – P(A)

The complement rule is a fundamental concept in probability theory. It provides a way to calculate the probability of an event not occurring, given the probability of the event occurring. In other words, it tells us the probability that the opposite of the event will happen.

The complement rule is expressed as follows:

P(A’) = 1 – P(A)

Where P(A’) is the probability of event A not occurring, and P(A) is the probability of event A occurring. The complement rule assumes that there are only two possible outcomes: A happening or A not happening.

The complement rule can be illustrated with an example:

Suppose you roll a fair six-sided die. The probability of rolling an even number (2, 4, or 6) is 1/2 (since there are three even numbers out of six possible outcomes). The complement of that event (rolling an odd number) is also 1/2, since there are three odd numbers out of six possible outcomes. Therefore, the complement rule tells us:

P(rolling an odd number) = 1 – P(rolling an even number)

or

P(rolling an odd number) = 1 – 1/2 = 1/2

Therefore, the probability of rolling an odd number is also 1/2, which is consistent with our intuition.

In summary, the complement rule is a useful tool for calculating probabilities of events that are hard to estimate directly. It can be used in a wide range of situations where there are only two possible outcomes, and where the probability of one outcome is known.

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