Complement of an Event (NOT) MathBitsNotebook.com Terms of Use   Contact Person: Donna Roberts
 If A is an event within the sample space S of an experiment, the complement of A (which may be written as AC) 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 stated:

 Experiment: Tossing a Coin Event A The coin shows heads. Complement AC The coin shows tails. Experiment: Drawing a Card The cards in a standard deck are either red or black. Event A The card is black Complement AC The card is red

 The probability of the complement of an event is one minus the probability of the event. Complement:

Since the sum of probabilities of all possible events equals 1, the probability that event A will not occur is equal to 1 minus the probability that event A will occur.

 A pair of dice are rolled. What is the probability of not rolling doubles? There are 6 ways to roll doubles: (1,1), (2,2), (3,3), (4,4), (5,5) (6,6). P(doubles) = 6/36 = 1/6 P(not doubles) = 1 - 1/6 = 5/6

 A pair of dice are rolled. What is the probability of rolling 10 or less? 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

 A gum ball machine contains gum balls of five different colors:  36 red, 44 white, 15 blue, 20 green, and 5 orange.  The machine dispenser randomly selects one gum ball.  What is the probability that the gum ball selected is: a.) green? b.) not green? c.) not orange? d.) orange? e.) red, white, or blue? f.) not a color in the flag of the USA? There are 120 gum balls 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.) the probability of red, white or blue is 36/120 + 44/120 + 15/120 = 95/120 = 19/24.  f.) find the complement of part e: 1 - 19/24 = 5/24.

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