Friday, August 10, 2012

A Very Easy Problem in Decision Theory

Honestly, I reckon a cat could do this one, mutatis mutandis:

A barrell contains 25% diamonds, and 75% circles.

10% of the diamonds are blue, the rest are red.

80% of the circles are blue, the rest are red.

One of the objects is pulled out at random, and you're told the colour.

You can take a guess at the shape. If you're right, then you get \$10, if you're wrong then you get a \$1 for your trouble.

You do pretty well if you always guess circle. Can you do better?

In fact it could be argued that a cat is a mechanism for solving such problems, amongst others.

1. This is probably another tricky math-logic thing that requires a meta view of Bayesian analysis or somesuch, but I don't come to the same conclusion.

Working through this with an example . . .

Assume 200 shapes in the barrel.
Therefore, 50 Diamonds (D) and 150 Circles (C).
Therefore, 5 Blue Diamonds (BD) and 45 Red Diamonds (RD).
Therefore, 120 Blue Circles (BC) and 30 Red Circles (RC).

Consequently, for Blue, 5BD and 120BC. Total Blue: 125.
Consequently, for Red, 45RD and 30RC. Total Red: 75.

Given that they tell me the color, and I am to guess the shape:

If they tell me Blue, I guess Circle. My odds of winning are 120/125, or 96%.

If they tell me Red, I guess Diamond. My odds of winning are 45/75, or 60%.

2. Looks good to me! Told you it was easy.