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MUTUALLY EXCLUSIVE EVENTS Two or more events are called mutually exclusive if the
occurrence of any one of them precludes the occurrence of any of the others.
The probability of occurrence of two or more mutually exclusive events is the
sum of the probabilities of the individual events. Sometimes when one event has occurred, the probability of
another event is excluded (referring to the same given occasion or trial). For
example, throwing a die once can yield a 5 or 6, but not both, in the same
toss. The probability that either a 5 or 6 occurs is the sum of their
individual probabilities.
EXAMPLE: From a bag containing 5 white balls, 2 black balls, and 11 red balls, 1
ball is drawn. What is the probability that it is either black or red? SOLUTION: The draw can be made in 18 ways. The choices are 2 black balls and 11
red balls, which are favorable, or a total of 13 favorable choices. Then, the
probability of success is
Since
drawing a red ball excludes the drawing of a black ball, and vice versa, the
two events are mutually exclusive; so the probability of drawing a black ball
is
and
the probability of drawing a red ball is
Therefore,
the probability of success is
EXAMPLE: What is the probability of drawing either a king, a queen, or a jack
from a deck of playing cards? SOLUTION: The individual probabilities are
Therefore,
the probability of success is
EXAMPLE: What is the probability of rolling a die twice and having a 5 and then
a 3 show or having a 2 and then a 4 show? SOLUTION: The probability of having a 5 and then a 3 show is
and
the probability of having a 2 and then a 4 show is
Then,
the probability of either p_{1} or p_{2} is
PRACTICE
PROBLEMS: When
tossing a coin, you have what probability of getting either a head or a tail? 2.
A bag contains 12 blue, 3 red, and 4 white marbles. What is the probability of
drawing a.
in 1 draw, either a red or a white marble? b.
in 1 draw, either a red, white, or blue marble? c.
in 2 draws, either a red marble followed by a blue marble or a red marble
followed by a red marble? 3.
What is the probability of getting a total of at least 10 points in rolling two
dice? (HINT: You want either a total of 10, 11, or 12.) ANSWERS:
