The Two Box Puzzle

The Puzzle

Someone presents you with two opaque boxes. One has two $100 bills. The other has a $100 bill and a $1 bill.

Box	Box
$100	$100
$1	$1

You are told that you will get to keep the second bill you withdraw.

You pull the first bill, and it’s $100. Should you take the second bill from the same box or the other one—or does it make no difference?

The Solution

There are two things you need to consider to answer this.

The box you already drew from is down one $100 bill. The other box is guaranteed to have at least one $100 bill, whereas the box you drew from might not have one at all.

Now it might seem like it’s better to switch boxes. But there’s one more consideration to make:

Before you pulled anything out, two of the three $100 bills were in the $200 box. When you pulled out a $100 bill, you got more information about the boxes: There is a 2 in 3 chance that the box you pulled from was the one with $200.

If you have not learned about conditional probability, it might be difficult to wrap your head around point 2. This blog post gives a simple introduction to the concept.

Let's put together everything we know.

If you pull from the same box:

There is a 2/3 chance it is the box with two $100 bills. If it turns out this way, there is a 100% chance of getting $100.
There is a 1/3 chance it is the box with both bills. If it turns out this way, there is a 0% chance of getting $100 since you already took out a $100 bill.

What are the odds of getting $100 if you pull from the same box?

2/3 x 100% + 1/3 x 0% = 2/3

There is a 2 in 3 chance of getting $100 if you pull from the same box.

If you pull from the other box:

There is a 2/3 chance it is the box with one of each bill. If it turns out this way, there is a 50% chance of getting $100.
There is a 1/3 chance it is the box with two $100 bills. If it turns out this way, there there is a 100% chance of getting $100.

What are the odds of getting $100 if you pull from the other box?

1/3 x 100% + 2/3 x 50% = 2/3

There is a 2 in 3 chance of winning if you pull from the other box.

Therefore, it doesn’t matter what you do!