Complementary Events

Definition 3.3

The complement of an event A is the event that A does not occur, that is, the event consisting of all sample points that are not in event A. We denote the complement of A by .

Here are some rule about probabilities:

The probability of an event happening is simply one minus the event not happening. That is, , or the probability of event A is one minus the probability of A not happening, (complement).

Therefore the sum of two complement events and is equal to 1:

If the events have no outcomes in common the probability of either of them happening is the sum of their probabilities. In notation, P (A or B) = P (A) + P (B).

Example. Observe the up face on a coin.

A ={Observe a head}, ={Observe a tail}.

, . Indeed, .

For example, suppose a certain little town the number of children in households with children is

Outcome 1 2 3 4 5 6 or moreProbability.15.55.10.10.05.05

The probability of two or fewer children is P (1 or 2)= P (1)+ P (2)=0.15+0.55=0.7.

Let’s denote A={1,2}. Then P()=0.7. How do you find P()?

= 1 - 0.7 = 0.3.

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