### Tip of the Week #33                     Tip Index

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# Answer to the "The Three Cards" Teaser Problem

Fun with statistics

Answer: 2/3. The side up is one of three equally likely outcomes:

1. the black side of the black/white card
2. side 'A' of the black/black card
3. side 'B' of the black/black card.

If the answer to this question was obvious, it would be much fun. Most people answer 1/2 to "The Three Cards" problem.

The point, of course, is to show that our intuitions about probabilities are often wrong, even about simple risk situations.

"The Three Cards" problem is similar to another that I've seen.

Suppose a cook has two pancakes. He says that one pancake is brown on both sides, and the other is brown on one side and golden on the other. He places one pancake on your plate, and it is brown on the side up. What is the probability that the other side is also brown?

Answer: 2/3. There are four possible outcomes for a pancake placed on your plate:

1. golden side up of brown/golden pancake
2. brown side up of brown/golden pancake
3. brown side up ('A') of brown/brown pancake
4. brown side up ('B') of brown/brown pancake

Again, most people erroneously judge that the pancakes are equally likely and the probability is 1/2. The new information, seeing a brown side up pancake on your plate, rules out possibility (1) of the four equally-likely outcomes. There is a 2/3 chance that the other side is also brown, outcomes (3) and (4).

—John Schuyler, January 1998.