### Tip of the Week #33
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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:

- the black side of the black/white card
- side 'A' of the black/black card
- 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:

- golden side up of brown/golden pancake
- brown side up of brown/golden pancake
- brown side up ('A') of brown/brown pancake
- 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.

Copyright © 1998 by John R. Schuyler. All rights reserved. Permission to copy with
reproduction of this notice.