elemprob-fall2010-page21

# elemprob-fall2010-page21 - own a dog Answer Let D be the...

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An example: A family has 2 children. Given that one of the children is a boy, what is the probability that the other child is also a boy? Answer. Let B be the event that one child is a boy, and A the event that both children are boys. The possibilities are bb,bg,gb,gg , each with probability 1 / 4. P ( A B ) = P ( bb ) = 1 / 4 and P ( B ) = P ( bb,bg,gb ) = 3 / 4. So the answer is 1 / 4 3 / 4 = 1 / 3. An example: Suppose the test for HIV is 98% accurate in both directions and 0.5% of the population is HIV positive. If someone tests positive, what is the probability they actually are HIV positive? Let D mean HIV positive, and T mean tests positive. P ( D | T ) = P ( D T ) P ( T ) = ( . 98)( . 005) ( . 98)( . 005) + ( . 02)( . 995) = 19 . 8% . 9 Bayes’ rule Suppose you know P ( E | F ) and you want P ( F | E ). An example: Suppose 36% of families own a dog, 30% of families own a cat, and 22% of the families that have a dog also have a cat. A family is chosen at random and found to have a cat. What is the probability they also
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Unformatted text preview: own a dog? Answer. Let D be the families that own a dog, and C the families that own a cat. We are given P ( D ) = . 36 , P ( C ) = . 30 , P ( C | D ) = . 22 We want to know P ( D | C ). We know P ( D | C ) = P ( D ∩ C ) / P ( C ). To ﬁnd the numerator, we use P ( D ∩ C ) = P ( C | D ) P ( D ) = ( . 22)( . 36) = . 0792. So P ( D | C ) = . 0792 /. 3 = . 264 = 26 . 4%. An example: Suppose 30% of the women in a class received an A on the test and 25% of the men received an A. The class is 60% women. Given that a person chosen at random received an A, what is the probability this person is a women? Answer. Let A be the event of receiving an A, W be the event of being a woman, and M the event of being a man. We are given P ( A | W ) = 21...
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