# Expressing intuition about surprise factor between three boxes in a guessing game

Suppose you are playing a guessing game where there is a box with three marbles. There are three kinds of boxes. Box A contains red, green and blue marbles; Box B contains red, green, and yellow marbles. Box C contains black, white and purple marbles. You are told that there is a 60% chance that this is Box A, 20% that it is Box B, and 20% that it is Box C.

Suppose you make a choice, and open the box. Although Box B and C are both equally unlikely, there is an intuitive sense in which Box B would be a less "surprising" choice than Box C, because it contains the red and green marbles (which are 80% likely to be encountered). How do I express this intuition probabilistically? Is the KL divergence between my marble color beliefs before and after opening the box smaller for Box B than Box C? How do I express this?

I know that P(red AND green AND yellow) = P(black AND white AND purple) = 20%.

• Expected surprise is entropy, see stats.stackexchange.com/questions/66186/… Commented Mar 17, 2023 at 14:25
• Expected surprise will be the same for Box B and Box C because they are equally likely, so this is not the measure I am looking for.
– Sam
Commented Mar 20, 2023 at 9:29