Complement of an Event
If A is an event within the sample space S of an activity or experiment, the complement of A (denoted A’) consists of all outcomes in S that are not in A.
The complement of A is everything else in the problem that is NOT in A.
Consider these experiments where an event and its complement are shown:
The probability of the complement of an event is one minus the probability of the event.
Complement:
P(A’) = 1 – P(A)
Example 1: A pair of dice are rolled. What is the probability of not rolling doubles?
Solution:
There are 6 ways to roll doubles.
P(doubles) = 6/36 = 1/6
P(not doubles) = 1 – 1/6 = 5/6
Example 2: A pair of dice are rolled. What is the probability of rolling 10 or less?
Solution: The complement of rolling “10 or less” is rolling 11 or 12.
P(10 or less) = 1 – P(11 or 12)
= 1 – [P(11) + P(12)] = 1 – (2/36 + 1/36) = 33/36 = 11/12
Example 3: A gumball machine contains gumballs of five different colors: 36 red, 44 white, 15 blue, 20 green, and 5 orange. The machine dispenser randomly selects one gumball. What is the probability that the gumball selected is:
(a) green?
(b) not green?
(c) not orange?
(d) orange?
(e) not a color in the flag of the USA?
(f) red, white, or blue?
Solution: There are 120 gumballs in total in the machine.
(a) the probability of green is 20/120 = 1/6.
(b) the probability of not green is 1 – 1/6 = 5/6.
(c) the probability of not orange is 1 – P(orange) = 1 – 5/120 = 1 – 1/24 = 23/24.
(d) the probability of orange is 1/24.
(e) find part f first and then use the complement. 1 – 19/24 = 5/24.
(f) the probability of red, white or blue is 36/120 + 44/120 + 15/120 = 95/120 = 19/24.