A red die is rolled. If the number rolled is no larger than 2 then a green die i

Question

A red die is rolled. If the number rolled is no larger than 2 then a green die is rolled twice, but if the number on the red die exceeds 2, the green die is rolled 3 times. Let A be the event that the number on the red die is no larger than 2. Let B be the event that 2 sixes are rolled on the green die. (That is, if the green die is rolled twice, both results are sixes; if it is rolled three times, the results are 2 sixes and 1 non-six, in any order.)
a) find the probability of event B, P(B)
b) find the probability of event A given event B, P(A|B)

Explanation / Answer

Given A is the event that the number on the red die is no larger than 2. and B is the event that 2 sixes are rolled on the green die
a. we have to find probability of event B, P(B) This is Bayes's theorem and total probability

probability of A: 2/6
A': 4/6

if A happens, probability of B is 1/36
if A does not happen: probability of B is 3 choose 2*1/36*5/6
second part follows the binomial probability

A intersect B=1/36*1/3
A complement intersect B=2/3*3 choose 2*1/36*5/6
P(B)=1/3*1/36+2/3*3*1/36*5/6=0.0555555556
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b)  probability of event A given event B
That's probabiliyt of A intersect B over probability of B
which is 2/6*1/36/P(B)=0.166666667

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