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An
event
is a set of outcomes. It is a subset of the sample space for an
activity or experiment.
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Event:
Drawing a black card from a deck of standard cards.
Probability of
this event = 26/52 = 1/2
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When an event
corresponds to a single outcome of the activity, it is often called a
simple
event.
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Simple
Event: Drawing the queen of spades from a deck of
standard cards.
Probability
of this event = 1/52 |
Two
events that have NO outcomes in common are called
mutually
exclusive.
These are events that cannot occur at the
same time.
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Mutually
exclusive - think of this as the 2 events together
(mutually) agreeing to exclude (not include) each others'
elements.
They have agreed to be different - mutually exclusive. |
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Examples:
A pair of
dice is rolled. The events of
rolling
a 6 and of
rolling
a double have
the outcome (3,3) in common. These two events are
NOT
mutually exclusive.
A pair
of dice is rolled. The events of
rolling
a 9 and of
rolling
a double have
NO outcomes in common. These two events
ARE
mutually exclusive. |
For
any two mutually exclusive events, the probability that an outcome will be
in one event or the other event is the sum of their individual
probabilities.
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If A
and B are mutually exclusive events,
P(A or B) = P(A) + P(B) |
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For
any two events which are not mutually exclusive, the probability that an
outcome will be in one event or the other event is the sum of their
individual probabilities
minus
the probability of the outcome being in both events.
Look
out!! Don't get stuck on this one!!!
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If
events A and B are NOT mutually
exclusive,
P(A or B) = P(A) + P(B) - P(A and B) |
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Example
1: A pair
of dice is rolled. What is the probability that the sum of the
numbers rolled is either 7 or 11?
Six
outcomes have a sum of 7:
(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)
P(7) = 6/36
Two
outcomes have a sum of 11:
(5,6), (6,5)
P(11) = 2/36
The
sum of the numbers cannot be 7 and 11 at the same time, so these events
are
mutually exclusive.
P(7
or 11) = P(7) + P(11) = 6/36 + 2/36 = 8/36 = 2/9
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Example
2: A pair
of dice is rolled. What is the probability that the sum of the
numbers rolled is either an even number or a multiple of 3?
Of
the 36 possible outcomes, 18 are even sums.
P(even) = 18/36 = 1/2
Sums
of 3, 6, 9, and 12 are multiples of 3.
There are 12 sums that are multiples of 3.
P(multiple of 3)= 12/36 = 1/3
However,
some of these outcomes appear in both events.
(not mutually exclusive)
The sums that are even and a multiple of 3 are 6 and 12.
There are 6 ordered pairs with these sums.
P(even AND a multiple of 3) = 6/36 = 1/6
P(even OR a multiple
of 3) = 18/36 + 12/36
- 6/36
= 24/36 = 2/3
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