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

This is tricky, because the specific wording of the question is super important here, and it’s probably where a lot of people are getting caught up on.

Let’s look at it this way: let’s say the first box has 1 $100 bill and 999 $1 bills (in other words, it would take forever to find the one $100 bill in it) and the second box is all $100 bills (in other words, it’s only possible to draw $100 bills from it).

You and your friend decide to take out bills for fun, to see if you can pull out a $100 bill. Your friend picks a box and draws a bill from it.

Nope, $1.

They put the bill back and the box back, mix around the boxes, pick a box again and draw a bill from it.

Nope, another $1.

They put the bill back and the box back, mix around the boxes, pick a box again and draw a bill from it.

Nope, another $1.

They put the bill back and the box back, mix around the boxes, pick a box again and draw a bill from it.

Oh hey, a $100 bill.

Then your friend turns to you and says, “I bet you dinner that the next bill I draw from this box is going to be a $100 bill.”

Would you take this bet?

I hope not, because it seems *reaaaaally* likely to me that he has the box with all $100 bills in it.

So this bet is in the form of “If I pick a random box **and** take out a $100 bill, what is the chance that the next bill I take from that box is another $100 bill”?

And the odds would be very very likely that you would draw another $100 bill. Much higher than 50%, even though there are only two boxes and only one has more than one $100 bill in it.

**But** if the bet is “If I pick a random box, what is the chance that I will draw one $100 bill, then draw another $100 bill?” then the odds are 50/50, because this will only happen if you pick the box with all the $100 bills in it, which is a 50% chance.

So back to your original question, with the very specific way it is worded, the probability is 2/3, because, since you drew the first $100 bill, you’ve demonstrated some evidence that the box you picked was a little more likely to be the one with two $100 bills in it. Like in the case where there were 1,000 bills in each box and one has 999 $1 bills, drawing a $100 bill from the box first gives you a little information about which box you might have.