If there are two boxes. The first has a 100$ bill and a 1$ bill, and the second has two 100$ bills. If I puck a random box and take out a 100$ bill, whaat is the chance of me taking out another 100$ bill?

Question

I’m honestly stuck. I’ve seen people say 1/2, others 2/3. Something Monty Hall Problem, Bayes Theorem but I’m still confused so here I am.

Edit: I believe you are not allowed to change your box choice on the 2nd “turn” as that would make having two boxes pointless, wouldn’t it?

In: 10

Answer 1

If you reach into a random box and take out a random bill, there are four possibilities, all equally likely. You will either grab the $1 bill, the $100 bill in the mixed box, the first $100 bill in the all-hundreds box, or the second $100 bill in the all-hundreds box. You grabbed a $100 bill, so we know the first possibility didn’t happen, but the other three are all still equally likely. So, there was a 1/3 chance you grabbed the $100 from the mixed box, a 1/3 chance you grabbed the first $100 bill in the all-hundreds box, and a 1/3 chance you grabbed the second $100 bill in the all-hundreds box. You will grab a $100 bill next if either of the second possibilities happened, so there is a 2/3 chance of getting another $100.

We can exaggerate the situation to make it more obvious. Let’s say the first box has a million bills in it, all of them $1 bills except for one $100 bill. The other box has a million $100 bills. If you reach into a random box and grab a $100 bill, you know that you almost certainly reached into the all-hundreds box, so the next bill you grab from that box will almost certainly be another $100, with a much better than 1/2 chance.