Question

A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.

Answer 1

Let the events of choosing the first and second bag be E₁ and E₂ respectively, then

P(E₁) = P(E₂) = `1/2`

Let E be the event of drawing red balls, then

P(E|E₁) = `4/8 = 1/2`, P(E|E₂) = `2/8 = 1/4`

Now, what is the probability of drawing a ball from the first bag when it is known that it is red in colour.

= P(E₁|E)

By Bayes' theorem P(E₁|E) = `(P(E_1) xx P(E|E_1))/(P(E_1) xx P(E|E_1) + P(E_2) xx P(E|E_2))`

= `(1/2 xx 1/2)/(1/2 xx 1/2 + 1/2 xx 1/4)`

= `(1/4)/(1/4 + 1/8)`

= `1/4 xx 8/3`

= `2/3`