Let U_1 and U_2 be two urns such that U_1 con | vedclass

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(NONE) Let $H$ be the event that the coin shows head and $T$ be the event that the coin shows tail. $P(H) = P(T) = 1/2$.Let $W$ be the event that the

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$23/30$

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(NONE) Let $H$ be the event that the coin shows head and $T$ be the event that the coin shows tail. $P(H) = P(T) = 1/2$.Let $W$ be the event that the ball drawn from $U_2$ is white.Case $1$: If $H$ occurs,$1$ ball is transferred from $U_1$ to $U_2$. $U_1$ has $3W, 2R$. The probability of transferring a white ball is $3/5$ and a red ball is $2/5$.If $W$ is transferred,$U_2$ now has $2W$. $P(W|H, 	ext{trans } W) = 1$.If $R$ is transferred,$U_2$ now has $1W, 1R$. $P(W|H, 	ext{trans } R) = 1/2$.$P(W|H) = (3/5 	imes 1) + (2/5 	imes 1/2) = 3/5 + 1/5 = 4/5$.Case $2$: If $T$ occurs,$2$ balls are transferred from $U_1$ to $U_2$. The possible pairs are $(2W), (1W, 1R), (0W, 2R)$.$P(2W) = \binom{3}{2} / \binom{5}{2} = 3/10$.$P(1W, 1R) = (\binom{3}{1} 	imes \binom{2}{1}) / \binom{5}{2} = 6/10$.$P(0W, 2R) = \binom{2}{2} / \binom{5}{2} = 1/10$.If $2W$ transferred,$U_2$ has $3W$. $P(W|T, 2W) = 1$.If $1W, 1R$ transferred,$U_2$ has $2W, 1R$. $P(W|T, 1W, 1R) = 2/3$.If $0W, 2R$ transferred,$U_2$ has $1W, 2R$. $P(W|T, 0W, 2R) = 1/3$.$P(W|T) = (3/10 	imes 1) + (6/10 	imes 2/3) + (1/10 	imes 1/3) = 3/10 + 4/10 + 1/30 = 7/10 + 1/30 = 22/30 = 11/15$.Using Bayes' Theorem: $P(H|W) = \frac{P(H)P(W|H)}{P(H)P(W|H) + P(T)P(W|T)} = \frac{1/2 	imes 4/5}{1/2 	imes 4/5 + 1/2 	imes 11/15} = \frac{2/5}{2/5 + 11/30} = \frac{12/30}{12/30 + 11/30} = 12/23$.

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