The probability entries are as follows: N means \"Nolan Arenadohas a good day\"

Question

The probability entries are as follows:

N means "Nolan Arenadohas a good day"

P(N) = 0.7

C means "Charlie Blackmonhas a good day"

P(C) = 0.4

L means "The Rockies lose"

P(L | (not N) and (not C)) = 0.8P

(L | (not N) and C) = 0.6

P(L | Nand (not C)) = 0.5

P(L | Nand C) = 0.2

B means "Bud Black is grumpy"

P(B| L) = 0.9

P(B| not L) = 0.2

M means "Mike is grumpy"

P(M | L) = 0.6

P(M| not L) = 0.3

S means "Stellais grumpy"

P(S|M) = 0.8

P(S| not M) = 0.1

You run into Bud in the evening and he is grumpy. What is the probability that both Nolan and Charlie had a good day today?

Explanation / Answer

Here, we are given that Bud Black is grumpy.

Now using the law of total probability, we get:

P(L) = P(L | (not N) and (not C))P((not N) and (not C)) + P(L | (not N) and ( C))P((not N) and ( C)) + P(L | ( N) and (not C))P(( N) and (not C)) + P(L | ( N) and ( C))P(( N) and ( C))

P(L) = 0.8*0.3*0.6 + 0.6*0.3*0.4 + 0.5*0.7*0.6 + 0.2*0.7*0.4 = 0.482

Now given that Bud Black is grumpy, using bayes theorem, probability that both Nolan and Charlie had a good day today is computed as:

P(N and C | B) = P(B | N and C) P(N)P(C) / P(B)

P(N and C | B) = 0.2*0.7*0.4 / 0.482 = 0.1162

Therefore 0.1162 is the required probability here.

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