Paragraph: Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 balls are drawn at random from…

Question

Paragraph: Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$ contains only 1 white ball. A fair coin is tossed. If head appears then 1 ball is drawn at random from $U_1$ and put into $U_2$. However, if tail appears then 2 balls are drawn at random from $U_1$ and put into $U_2$. Now, 1 ball is drawn at random from $U_2$. Question: Given that the drawn ball from $U_2$ is white, the probability that head appeared on the coin is

Aicharya · Accepted Answer

$P$ (head appeared/white from $V_2$ ) $ \begin{aligned} & =P(H) \cdot \frac{\left\{\frac{{ }^3 C_1}{{ }^5 C_1} 	imes \frac{{ }^2 C_1}{{ }^2 C_1}+\frac{{ }^2 C_1}{{ }^5 C_1} 	imes \frac{{ }^1 C_1}{{ }^2 C_1}ight\}}{\frac{23}{30}} \ & =\frac{1}{2} \frac{\left\{\frac{3}{5} 	imes 1+\frac{2}{5} 	imes \frac{1}{2}ight\}}{\frac{23}{30}}=\frac{12}{23} \ & \end{aligned} $

Paragraph: Let $U_1$ and $U_2$ be two urns such that $U_1$ contains 3 white and 2 red balls and $U_2$…

Solution