Can someone help me with the following scenario, found on the Wikipedia page on Bayesian Inference:
Suppose there are two full bowls of cookies. Bowl #1 has 10 chocolate chip and 30 plain cookies, while bowl #2 has 20 of each. Our friend Fred picks a bowl at random, and then picks a cookie at random. We may assume there is no reason to believe Fred treats one bowl differently from another, likewise for the cookies. The cookie turns out to be a plain one. How probable is it that Fred picked it out of bowl #1?
I understand how to work out the probability when Fred picks one bowl and one cookie at random, which is:
\begin{align} P(H_1|E) &= \frac{P(E|H_1)\,P(H_1)}{P(E|H_1)\,P(H_1)\;+\;P(E|H_2)\,P(H_2)} \end{align}
\begin{align} = \frac{0.75 \times 0.5}{0.75 \times 0.5 + 0.5 \times 0.5} \ \ \ & = 0.6 \end{align}
What I would like to understand is how to work out the probability if Fred picks up a second cookie from the same bowl.
Can anyone help me with the formula?
Thanks