Dice Games and Children

Enough of ruminations about ideal tax cheats. When learned papers are written trying to explain why people comply with the tax code, who am I to try to explain it meaningfully in 500 words or less? If you want to delve deeper into the issue, check out the reference I gave or try searching in key words.

Today I want to revert to a very old puzzle–game actually–that has variations beings played to this day. Sometimes it is played with three dice and sometimes on a large wheel which has all possible combinations of three dice, so the two variations are identical for our purposes.

The game is posed to you as a fair one. You bet $1 on a number. The dice are thrown, or the wheel spun. If your number comes up on one die, you get $2 (your original and one payoff). If it comes up on two dice, you get $3. If all three come up your number, you collect $4. Alas, if your number does not come up at all, you lose your dollar.

If you think about it for a minute, you see that the probability of any die showing your number is 1/6, but since three of them are thrown at once, the probability of winning must be 3(1/6) or ½. The odds are even and it is a fair game.

But upon further reflection, you realize the fancy casino and glittering wheel must be paid for somehow, or the grubby guy in the alcove with three dice must be there for a reason.

The house probably has an advantage. What is it?

This practical analysis occurs many times in making decisions (in this case whether to gamble or not). The seductive easy analysis overlooks the difference between linked and independent events. Similar faulty reasoning can lead to inaccurate predictions when many independent variables are in play.

Computer codes can correctly handle problems of this type, but humans seem to have difficulty with them. This combined with our general inability to correctly estimate probabilities of individual components often leads to incorrect predictions. That is why gaming problems based on simple models such as a 6-side die are valuable. One can intuitively see what the probability of each isolated event is.

The difficulty comes in understanding the effects of many uncorrelated events.

After you solve the dice problem, consider that I have two children. At least one of them is a boy. What is the probability the other is a boy also? Assume the boy/girl ratio in the population at large is 50/50 and that both children are adopted so genetics do not enter.

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