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?

#### Solution

Let event J_{1}: Ball drawn from jar I,

event J_{2}: Ball drawn from jar II.

P(J_{1}) = P(head) = `1/2`

P(J_{2}) = 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 condtion that it is drawn from Jar I.

= `"P"("W"/"J"_1)`

= `(""^5"C"_1)/(""^12"C"_1)`

= `5/12`

Similarly, `"P"("W"/"J"_2)`

= `(""^3"C"_1)/(""^15"C"_1)`

= `3/15`

= `1/5`

Required probability = `"P"("J"_2/"W")`

By Bayes’ theorem

`"P"("J"_2/"W") = ("P"("J"_2) "P"("W"/"J"_2))/("P"("J"_1) "P"("W"/"J"_1) + "P"("J"_2) "P"("W"/"J"_2))`

= `(1/2 xx 1/5)/(1/2 xx 5/12 + 1/2 xx 1/5)`

= `(1/5)/((25 + 12)/60`

= `12/37`.