1. 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?

p1 p2 3/8

3/8 cookies

To determine the probability that Fred picked the plain cookie from Bowl #1, we can use Bayes' theorem. Let's break it down step by step:

Step 1: Determine the total number of cookies in both bowls.
- Bowl #1: 10 chocolate chip + 30 plain = 40 cookies
- Bowl #2: 20 chocolate chip + 20 plain = 40 cookies

Step 2: Calculate the probability of Fred picking each bowl.
- Since Fred randomly picks a bowl, the probability of picking Bowl #1 is 1/2 (since there are two bowls to choose from) and the probability of picking Bowl #2 is also 1/2.

Step 3: Calculate the probability of picking a plain cookie from each bowl.
- In Bowl #1, there are 30 plain cookies out of 40, so the probability of picking a plain cookie from Bowl #1 is 30/40 = 3/4.
- In Bowl #2, there are 20 plain cookies out of 40, so the probability of picking a plain cookie from Bowl #2 is 20/40 = 1/2.

Step 4: Calculate the combined probability of picking a plain cookie from either bowl.
- Since Fred randomly picked a bowl in Step 2, we need to consider both possibilities. So, we calculate the weighted average combining the probabilities obtained in Step 3 using the probabilities obtained in Step 2:
- Probability of picking a plain cookie = (Probability of picking Bowl #1) * (Probability of picking a plain cookie from Bowl #1) + (Probability of picking Bowl #2) * (Probability of picking a plain cookie from Bowl #2)
- = (1/2) * (3/4) + (1/2) * (1/2)
- = 3/8 + 1/4
- = 3/8 + 2/8
- = 5/8

Therefore, there is a 5/8 or 62.5% probability that Fred picked the plain cookie from Bowl #1.

To get a plain cookie:

we could have (1,P) or (2,P)
prob (1, P)= (1/2)((30/40) = 3/8
prob (2,P) = (1/2)(20/40) = 1/4

prob(plain cookie) = 3/8 + 1/4 = 5/8

(checking:
prob(1,C) = (1/2)(10/40) = 1/8
prob(2,C) = (1/2)(20/40) = 1/4
note 3/8+1/4+1/8+1/= 1 , check! )

could have stopped at
prob (1, P)= (1/2)((30/40) = 3/8