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JEE Advance - Mathematics (2011 - Paper 1 Offline - No. 12)

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 $${U_2}$$ being white is
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 $${U_2}$$ being white is
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 $${U_2}$$ being white is
The probability of the drawn ball from $${U_2}$$ being white is
$${{13} \over {30}}$$
$${{23} \over {30}}$$
$${{19} \over {30}}$$
$${{11} \over {30}}$$

Explanation

H $$\to$$ One ball from U1 to U2.

T $$\to$$ Two balls from U1 to U2.

E : One ball drawn from U2.

P/W from

$${U_2} = {1 \over 2} \times \left( {{3 \over 5} \times 1} \right) + {1 \over 2} \times \left( {{2 \over 5} \times {1 \over 2}} \right) + {1 \over 2} \times \left( {{{{}^3{C_2}} \over {{}^5{C_2}}} \times 1} \right) + {1 \over 2} \times \left( {{{{}^2{C_2}} \over {{}^5{C_2}}} \times {1 \over 3}} \right) + {1 \over 2} \times \left( {{{{}^3{C_1}\,.\,{}^2{C_1}} \over {{}^5{C_2}}} \times {2 \over 3}} \right) = {{23} \over {30}}$$

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