Question

A game is played by first flipping a fair coin, then rolling a die multiple times. If the coin lands heads, then die A is to be used; if the coin lands tails, then die B is to be used. Die A has 4 red and 2 white faces, whereas die B has 2 red and 4 white faces. If the first two throws result in red, what is the probability that the coin landed on heads?

Answer #1

Table below denotes all possible outcomes:

Let N denotes the event that red shows up in the first n throws, then using Bayes formula, we have

P(H|N) = P(H) P(H|N) / P(N) = (1/2)(2/3)^n / [(1/2)((1/3)^n + (2/3)^n)]

= 2^n / (1+2^n)

Now if the first two throws result in red, the probability that the coin landed on heads can simply be obtained by putting n=2 in above probability of first n throws.

Therefore the required probability that the coin landed on heads when the first two throws result in red is given by:

P(H | N_{n=2}) = 2^2 / (1 + 2^2) = 4 / (1+4) = 4/5
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