Two Boxes

When you pull out a $100 bill, it’s twice as likely to have come from the $200 box as from the $101 box, so there’s a 2/3 chance you pulled from the $200 box and a 1/3 chance you pulled from the $101 box. (If you struggle to understand this, try the Gold and Silver Puzzle first.)

If you stay with 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.

2/3 × 1 + 1/3 × 0 = 2/3

If you switch boxes:

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 is a 100% chance of getting $100.

2/3 × 1/2 + 1/3 × 1 = 2/3

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