It is known that for Freshman at a certain university 68 are

It is known that for Freshman at a certain university, 68% are taking a math class, 54% are taking a history class, and 35% are taking both courses. What is the probability that a randomly selected freshman at this university is

a) taking neither a math class nor a history class?

b) taking a math class, but not a history class?

c) is taking only history given he is taking at least one of the two classes?

Solution

Given

P(math)=0.68

P(history)=0.54

P(both) =0.35

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a) taking neither a math class nor a history class?

1-P(math or history)

=1-P(math) - P(history) + P(both)

=1-0.68 -0.54+0.35

=0.13

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b) taking a math class, but not a history class?

P(math but not history)

=P(math) - P(both)

=0.68-0.35

=0.33

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c) is taking only history given he is taking at least one of the two classes?

P( only history | at least one classes)

= P(only history)/P(at least one classes)

=(0.54-0.35)/(0.68 +0.54-0.35)

=0.2183908