# Jar I contains 5 white and 7 black balls. Jar II contains 3 white and 12 black balls. A fair coin is flipped; if it is Head, a ball is drawn

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Jar I contains 5 white and 7 black balls. Jar II contains 3 white and 12 black balls. A fair coin is flipped; if it is Head, a ball is drawn from Jar I, and if it is Tail, a ball is drawn from Jar II. Suppose that this experiment is done and a white ball was drawn. What is the probability that this ball was in fact taken from Jar II?

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Let event J1 : Ball drawn from jar I,

event J2 : Ball drawn from jar II.

P(J1) = P(head) = 1/2

P(J2) = P(tail) = 1/2

Let event W: Ball drawn is white.

In Jar I, there are total 12 balls, out of which 5 balls are white.

∴ Probability that the ball drawn is white under the condition that it is drawn from Jar I.

P(W/J1) = $\frac {^5C_1}{^{12}C_1} = \frac {5} {12}$

Similarly, P(W/J2) = $\frac {^3C_1}{^{15}C_1} = \frac {3} {15} = \frac 1 5$

Required probability = P(J2/W)

By Bayes’ theorem 