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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:
|
Experiment:
Tossing a coin |
| Event |
A |
The coin shows
heads. |
| Complement |
A' |
The coin shows
tails. |
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Experiment:
Drawing a card |
| Event |
A |
The card is black. |
| Complement |
A' |
The card is red. |
The
probability of the complement of an event is one minus the probability of
the event.
|
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Complement:
P(A')
= 1 - P(A) |
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Example
1: A
pair of dice are rolled. What is the probability of not rolling
doubles?
There are 6 ways to
roll doubles.
P(doubles)
= 6/36 = 1/6
P(not doubles) = 1 - 1/6 = 5/6 |
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Example
2: 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 |
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? |
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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. |
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