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# Complementary Events

Complementary events are those events where the probability of one event precludes the happening of the other event, say when you toss a coin, you get heads, but not tails. So getting heads or tails are complementary events.

The sum of probabilities of all possible events is always one. When you toss a coin, the probability of getting heads is 0.5, and the probability of getting tails is 0.5 and the sum of these two events is 1.

So if we are given that the probability of getting heads P (Heads) = , then

Probability of getting tails P (Tails) = 1 – P (Heads)

=  1 – =  1 – 0.5  =  0.5

Another way of looking at this situation will be to think that when you get heads, you have not got tails. Or when you get tails, you have not got heads. Hence heads (or tails) and not getting heads (or not getting tails) are called complementary events. These events are opposite to each other’s occurrence.

Similarly winning a game and not winning a game are complementary events, and the sum of their probabilities will always be 1.

If P (winning)  = ,

P (not winning)  =  1 – P (winning)

= 1 – = If the probability of an event occurring is P(E), then the probability of the event not occurring is represented as P( ) or P(E), where

P(E)  +  P (not E)  =  P(E)  +  P( )  =  1.

(E) and ( ) are complementary events.

Examples:

1. A bag has 2 green, 4 orange, 1 purple and 3 red balls. The probability of picking a red ball P(red) is .

So the probability of not picking a red ball P(not red) = .

P(red) + P(not red)  = + =  1.

2. Luke’s chance of clearing the high jump is . What is his chance of not clearing the high jump?

Here either Luke clears the high jump, or he doesn’t. So these two events are complementary events.

P(not clearing)  =  1 – P(clearing)

=  1 – = Summary

If all outcomes are equally likely:

Probability = P(E)  = where

n(E) = number of favourable outcomes

n(S) = number of possible outcomes

Total of all probability is 1. Therefore

P(E) + =  1, or = 1 – P(E), where

P(E) is the probability of the original event, and is the probability of the complementary event.