Kiran has two bags. Bag I has 7 red and 2 blue balls and bag II has 5 red and 9 blue balls. Amy draws a ball at random and it turns out to be red. Find the probability that the ball was from the bag I using the Bayes theorem.

Let X and Y be the events that the ball is from the bag I and bag II, respectively.

probability of chossing a bag for drawing a ball

P(X) = P(Y) = \$\frac(1)(2)\$

probability of drawing a red ball from bag 1

P(A/X) = \$\frac(7)(11)\$

probability of drawing a red ball from bag 2

P(A/Y) = \$\frac(5)(14)\$

Bayes theorem \$(P(E_1/A) = P(E_1) * P(A/E_1)) / (P(E_1) * P(A/E_1) + P(E_2) * P(A/E_2))\$

Simplification

\$\frac(7)(11) \times \frac(1)(2) / \frac(7)(11) \times \frac(1)(2) + \frac(5)(14) \times \frac(1)(2)\$

After simplification we get

0.64

Answer

C