Should I pick the higher dice?

Question

Assuming I start with $n$ dice that have been rolled once, is it beneficial to choose the higher dice when I roll less than $n$ dice again (assuming I want a high roll)?

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In some board games, dice play a big part. And depending on the circumstance, a different amount of dice is used. In this particular case, I'm assuming that the higher the resulting number, the better. Also, I'm assuming not perfectly fair dice, since those rarely happen in everyday play.

Here's my logic: If I roll 7 dice, and 4 are high (5/6) while 3 are low (1/2), should I pick the high dice if I only need to roll 4? (Assuming that this is the first time I see the dice rolled).

This seems as if it might make sense because if the dice aren't fair and the only result we've seen so far is a higher number, it might make it seem as if that number is more likely to occur.

On the other hand, this almost seems like the gambler's fallacy. Except that fallacy assumes perfect dice.

So should I pick the higher dice? Or does it really not matter? If it does matter, by how much?

Answer 1

Why wouldn't you? If the dice are fair, you lose nothing. If they are not fair, you have some advantage. The only way you lose is if successive rolls are anti-correlated.

That said, I suspect if you start keeping careful records for your friend's rolls, there will be no advantage. Selective memory is very powerful. People are excellent at finding patterns, even when they are not there.

Answer 2

If I understand your question correctly:

Suppose we have $n$ weighted dice, with unknown weights.
We roll them once, and are tasked with selecting $k$ with the highest expected value on a second roll.

Let us imagine for simplicity two coins.

Coin A is weighted at 75% heads, 25% tails.
Coin B is weighted at 25% heads, 75% tails.

We wish to, given only knowledge from the first flip, select coin A.

Flipping both of them to start, we have three outcomes:
$\dfrac34\dfrac34 = \dfrac 9{16}$ chance, A heads, B tails. Choose correctly.
$\dfrac34\dfrac14 = \dfrac 3{16}$ chance, A heads, B heads. Gain no information.
$\dfrac14\dfrac14 = \dfrac 1{16}$ chance, A tails, B heads. Choose incorrectly.
$\dfrac34\dfrac14 = \dfrac 3{16}$ chance, A tails, B tails. Gain no information.

From this quick thought experiment, we see that this strategy- choosing the dice with the highest number- will more often yield better results.
Similar analysis can be performed on standard 6-sided dice, but is lengthy to write out for little added benefit.

However, modern dice are manufactured and tested to a high degree, and use multiple factors to try to be as fair as possible (notice how on nearly all dice with an even number of sides, opposite sides sum to the same number.) Thus, unless your friend is cheating with actual weighted dice, it should not matter in practice.

If your friend is cheating, I suggest playing board games with someone more fun and fair; your problem is not in the dice.

Answer 3

If you know a die is biased to giving you 6's (and otherwise equal), you would roll it, correct?

Each roll of a die gives you some information about the actual likelihood of rolling different numbers on that die. In principle you could use that information to decide which dice to use.

In practice, it would take hundreds or thousands (or hundreds of thousands?) of die rolls to really get a read on the actual distribution of the die, and carefully tabulating how many times each side comes up.

A human being simply can't "get a feel for the die" by rolling it a few times, unless the weighting is very, very uneven.