# Box A contains 2 black and 3 red balls, while box B contains 3 black and 4 red balls.

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Box A contains 2 black and 3 red balls, while box B contains 3 black and 4 red balls. Out of these two boxes one is selected at random; and the probability of choosing box A is double that of box B. If a red ball is drawn from the selected box, then find the probability that it has come from box B.

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Let the events be defined as:

A: Selection of box A

B: Selection of box B

R: Drawing a red ball.

Let P (B) = p. Then, according to given condition P(A) = 2P (B) = 2p

$P(\frac{R}{A}) =\frac{^3C_1}{^5C_1}= \frac{3}{5}$ ,$P(\frac{R}{B}) =\frac{^4C_1}{^7C_1}= \frac{4}{7}$

∴ $P(\frac{B}{R}) = \frac{P(B).P(\frac{R}{B})}{P(A).P(\frac{R}{A})+P(B).P(\frac{R}{B})}$ = $\frac{p.\frac{4}{7}} {2p.\frac{3}{5}+p.\frac{4}{7}}$ =$\frac{4/7}{6/5+4/7}$ = $\frac{4/7}{\frac{42+20}{35}} =\frac{4/7}{62/35}$

$\frac{10}{31}$