The complement event. How to calculate the probability of not A.

The probability of an event not happening is known as the complement event. To work out the complement of an event subtract the probability from 1. This can be summarised by the following formula:

P(not A) = 1 – P(A)

Let's look at a few examples of working out the complement of A.

Example 1

Cheryl is about to take her driving test. The probability that Cheryl passes the driving test is 0.65. Work out the probability that she fails the driving test?

All you need to do is subtract the probability of passing the test from 1:

P(not passing) = 1 – P(passing)

P(not passing) = 1 – 0.65

P(not passing) = 0.35

Example 2

The probability of spinning the colour red on a spinner is 3/7. Work out the probability of not spinning a red on the spinner.

All you need to do is subtract the probability of spinning a red away from 1:

P(not red) = 1 – P(red)

P(not red) = 1 – 3/7

P(not red) = 4/7

Example 3

Work out the probability of not rolling a 5 on a fair dice.

Since you know the probability of rolling a 5 is 1/6 then you need to subtract this from 1 to get the compliment event:

P(not 5) = 1 – P(5)

P(not 5) = 1 – 1/6

P(not 5) = 5/6

Example 4

If the probability that it rains tomorrow is 0.18, find the probability that tomorrow will be dry.

Here you need to work out the probability that it doesn’t rain. Like the last three examples, subtract the probability of it raining from 1:

P(not rain) = 1 – P(rain)

P(not rain) = 1 – 0.18

P(not rain) = 0.82

Example 5

On a spinner there are 3 colours; red, blue and green. The probability of spinning a red is 0.14 and the probability of spinning a blue is 0.35. Work out the probability of spinning a green.

First, you need to work out the probability of spinning a red or blue which can be done by adding 0.14 and 0.35 together:

0.14 + 0.35 = 0.49

All you need to do now is subtract the probability from 1 to give the probability of not spinning a red or blue:

P(not a red or blue) = 1 – P(red or blue)

= 1 – 0.49

= 0.51

So the probability of spinning a green is 0.51.

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