# Given that event e has a probability of 0.31, the probability of the complement of event e?

**Solution:**

If P(e) is the probability of the event ‘e’ happening, then the complement of event ‘e’ implies event NOT HAPPENING and that is denoted by P(\(\overline{e}\)).

P(\(\overline{e}\)) = 1 - P(e) --- (1)

So if the probability of event “e” happening is 0.31 then the complement of event ‘e’ is :

P(\(\overline{e}\)) = 1 - 0.31 = 0.69

Now, what is the significance of a compliment? Actually many times it is easy to find the probability of the complement of an event rather than the event itself. And from its complement, the probability of the event can be determined by using an equation i.e.

P(e) = 1 - P(\(\overline{e}\))

Sometimes it is cumbersome to determine the probability of an event, but it is far easier to determine the probability of its complement. Let us illustrate the statements:

**Example.**

Suppose there is an experiment of rolling two fair dice. The event for which the probability has to be determined is A = sum of outcomes of the two dice is at least 3.

This means that the probability of the sum being either 3, or 4, or 5, or 6, or 7, or 8, or, 9, or 10, or 11, or 12 is to be determined i.e.

P(A) = P(sum of 3 on two dice) + P(sum of 4 on two dice) + ……+ P(sum of 12 on two dice)

This is a cumbersome process, instead, the probability of the complement of A is calculated. The complement of A will be the probability of determining the occurrence of the event of the sum of 2 on the rolling of the two dice.

P( \(\overline{A}\)) = (1/6)(1/6) = 1/36

P(A) = 1- P( \(\overline{A}\))

= 1 - 1/36 = 35/36

## Given that event e has a probability of 0.31, the probability of the complement of event e?

**Summary:**

Given that event e has a probability of 0.31, the probability of the complement of event e is 1 - 0.31 = 0.69.

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