1)Use Bayes\' theorem or a tree diagram to calculate the indicated probability.
ID: 3142320 • Letter: 1
Question
1)Use Bayes' theorem or a tree diagram to calculate the indicated probability. Round your answer to four decimal places.
P(A | B) = .2, P(B) = .8, P(A | B') = .1. Find P(B | A).
P(B | A) =
2) Suppose that it snows in Greenland an average of once every 20 days, and when it does, glaciers have a 23% chance of growing. When it does not snow in Greenland, glaciers have only a 2% chance of growing. What is the probability that it is snowing in Greenland when glaciers are growing? (Round your answer to four decimal places.)
Explanation / Answer
1. P(A|B) = .6 P(B) = .4
P(A|B) = P(AB) / P(B)
=> .6 = P(AB) / .4
=> P(AB) = .6 * .4 = .24
P(A|B') = .8
=> P(AB') / P(B') = .8
=> [P(A) - P(AB)] / ( 1 - P(B)) = .8
=> [P(A) - .24] / (1 - .4) = .8
=> [P(A) - .24] / .6 = .8
=> P(A) - .24 = .8 * .6 = .48
=> P(A) = .48 + .24 = .72
Using Baye's theorem,
P(B|A) = P(B) / P(A) * P(A|B)
=> P(B|A) = .4 / .72 * .6
=> P(B|A) = 0.3333
2. Let A: Snowing and B: Glaciers growing
Given P(A) = 1/20 = 0.05
and
P(B|A) = 0.23
P(B|A) = P(AB) / P(A)
=> .23 = P(AB) / .05
=> P(AB) = .23 * .05 = 0.0115
Also
P(B|A') = 0.04
=> P(BA') / P(A') = .8
=> [P(B) - P(AB)] / ( 1 - P(A)) = .8
=> [P(B) - 0.0115] / (1 - 0.05 ) = .8
=> [P(B) - 0.0115] / 0.95 = 0.8
=> [P(B) - 0.0115] = 0.8 * 0.95 = 0.76
=> P(B) = 0.76 + 0.0115 = 0.7715
Using Baye's theorem,
P(A|B) = P(A) / P(B) * P(B|A)
= 0.05 / 0.7715 * 0.23
= 0.0149
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