# Unit 21 Section 4 : Complementary Events

Two events are described as complementary if they are the only two possible outcomes.

For example, imagine we are testing whether it rains on a particular day.
The events "it rains" and "it doesn't rain" are complementary because:
- only one of the two events can occur
- no other event can occur
Therefore, these two events are complementary.

For another example, consider the rolling of a die to see whether the result is odd or even.
The events "odd" and "even" are complementary because:
- the result must be either "odd" or "even" (not both)
- the result cannot be anything except "odd" or "even"
Therefore, these two events are also complementary.

Finally, consider a bag only containing 4 white balls and 5 black balls.
We are interested in whether a ball picked from the bag is white or black.
The events "white" and "black" are complementary.
The probability of "white" is p(white) = . The probability of "black" is p(black) = .
Notice how p(white) + p(black) = 1.

 If A is an event, and A' is the complementary event, p(A) + p(A') = 1 or p(A') = 1 - p(A)

So, to find the probability an event not happening, we need to subtract the probability of that event from 1.

Practice Questions
Work out the answers to these questions then click to see whether you are correct.

 (a) The probability of Jane winning her tennis match is 5 . 7
What is the probability of her not winning the match? (b) A spinner only has whole numbers as outcomes. The probability of getting an even number is 0.27.
What is the probability of getting an odd number? ## Exercises

Work out the answers to the questions below and fill in the boxes. Click on the button to find out whether you have answered correctly. If you are right then will appear and you should move on to the next question. If appears then your answer is wrong. Click on to clear your original answer and have another go. If you can't work out the right answer then click on to see the answer.

 Some of the answers in this section are fractions. Each fraction has two input boxes. Put the numerator in the top box and the denominator in the bottom box, like this: In this exercise, you must also simplify any fractions in your answers.
Question 1
 The probability that Scott will win his next darts match is 3 . 5
What is the probability that he will not win?   Question 2
 The probability that it will snow on Christmas Day is 1 . 8
What is the probability that it will not snow on Christmas Day?   Question 3
 The probability that a child speaks some French is 7 . 20
What is the probability that a child does not speak some French?   Question 4
The probability that Natasha is late for school is 0.1.

What is the probability that she is not late? (give your answer as a decimal)  Question 5
The probability that Sergio gets all his spellings correct in his next test is 0.75.

What is the probability that he does not get them all correct? (give your answer as a decimal)  Question 6
 If you take a card at random from a pack of playing cards, the probability of getting a king is 1 . 13
What is the probability that you do not get a king?   Question 7
One of the numbers in the Venn Diagram below is chosen at random. (a) What is the probability that the number is in the set A?   (b) What is the probability that the number is not in the set A?   Question 8
A bag contains 100 balls, each marked with a number from 1 to 100. A ball is taken from the bag at random.

(a) What is the probability that the number on the ball is a multiple of 3?   (b) What is the probability that the number on the ball is not a multiple of 3?   You have now completed Unit 21 Section 4
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