Question

Say I have a wallet that contained either a $2 bill or a $20 bill (with equal likelihood), but I don’t know which one. I add a $2 bill. Later, I reach into my wallet without looking and remove a bill. It’s a $2 bill. There’s one bill remaining in the wallet. What is the chances that it’s a $2 bill?

Answer 1

Answer:

        $\large\boxed{\large\boxed{2/3}}$

Explanation:

The initial scenary is:

• One $2-bill with 50% probability
• One $20-bill with 50% probability

I will call the original $2-bill $2a and the $2-bill that I add $2b-bill.

When $2b-bill is added the options are:

• One $2a-bill plus $2b-bill with 50% probability
• One $2b-bill plus one $20-bill 50% probability

When I remove one of the $2 bills there are three possibilities for the bills on the wallet:

• One $2a-bill,
• One $2b-bill, and
• One $20-bill

Each option is equally likely. Then each option has probability 1/3.

Two of those options are a $2 bill ($2a-bill and $2b-bill).

Hence, the probability of the bill remaining in the wallet be a $2 bill is 1/3 + 1/3 = 2/3.