# Probability of Dice Rolls Spread across Rollers

There’s a coffee shop I frequent that has a game with five 6-sided dice. If you roll four of a kind, you get a free drink. If you roll five of a kind, you get 10 free drinks. (I’ve rolled four of a kind twice.)
The odds of rolling four of a kind are ~1/52. The odds of rolling five of a kind are ~1/1,296. Does this mean that I can expect a four of a kind once every 52 or so rolls and a five of a kind every 1,296 or so rolls OR does it mean that the dice will come up four of a kind every 52 or so rolls and five of a kind every 1,296 or so rolls?
It seems to me the probability isn’t connected to the roller but rather to the dice, therefore the latter. But aren’t odds also expressed the first way? Do the separate statements (“I will roll four of a kind about every 52 rolls” and “The dice will come up four of a kind about every 52 rolls”) work out to be the same thing? I may have confused myself.

Different dice rolls should be stochastically independent, so it does not matter to you if other people have been rolling the dice.

Yes, but aren’t odds often expressed with the bettor as the subject? “Your odds of winning are…”; “The odds of you being eaten by a shark are…”
In the game described, what then are the odds of me rolling four of a kind or five of a kind, on any given roll?

Odds aren’t calculated with respect to a ‘roller’. You often hear ‘your odds are…’ because the common situation is that the odds are being described to an individual. But it makes no difference if one person rolls a die ten times or ten people each roll it once. It’s still ten rolls.

Also, the odds do not change after any arbitrary number of rolls. If you are trying for a six you have a 1/6 chance of hitting it. If you roll five times and don’t get a six, the odds of you getting a six on the next roll are still 1/6. The trials are completely independent. Assuming the dice are fair and balanced, no number is ever ‘due’, and dice do not run ‘hot’ or ‘cold’.

So what are the odds of me rolling five of a kind with five 6-sided dice?

The probability of rolling five of a kind is 1/1296, regardless of who is doing the rolling, as long as it’s done “fairly” (all possible dice rolls equally likely to appear).

1/6^4

I don’t believe some outcome can be “due”, and if some outcome is “hot” that indicates to me the game is fixed.

“Once is happenstance, twice is coincidence, three times is enemy action.”

Very wise.

The notion that something is “due” to happen because it hasn’t happened in a while is known as the “Gambler’s Fallacy”—at least in a situation, like dice rolling, where the outcomes are all supposed to be independent of one another.

If they’re not independent, it’s not necessarily a fallacy. For example, suppose I got a deck of cards, shuffled it, and started turning cards over one by one. If I got well over halfway through the deck without turning up any face cards, it would be reasonable to think that a face card was “due” and that the probability of the next card being a face card was significantly higher than it had been at the beginning.

In general, the probability of event A occurring is (Number of ways for event A to happen) / (Total number of outcomes). In the case of five of a kind in dice that works out to 6/7776 or 1/1296.

It means that if you roll five dice N times (where N is some very large number), about x/52 of those trials will yield four of a kind, where x is a number close to 1. If you roll five dice N times, about x/1296 of those trials will yield five of a kind. There’s no guarantee your defined dice events will occur every 216 or 1296 rolls. A flipped coin has odds of landing heads of 1/2, but you can have it land on heads multiple times in a row.

If I pick any N rolls, in any way that isn’t based on the outcome of the rolls, then on average N/52 rolls will be a four of a kind. This could be the first N rolls ever made in the coffee shop, it could be the first N rolls you ever made, it could be the first roll made each day in the shop for the next N days, it could be every other roll until there are N of them, whatever. It doesn’t matter if you change who’s rolling them, or you replace all of the physical dice, or if you go to a different coffee shop that offers the same game.

Although, if you roll 5 of a kind and are asked to bet on which subsequent roll you will achieve the feat again, the smart bet is on the next roll.

Why is the next roll more likely to be five-of-a-kind than any subsequent roll?

Because there’s a non-zero chance the dice are loaded, and you want to cash in before the game is shut down because of it.

It’s unlikely enough you suspect trick dice, i.e. the fair dice and independence assumptions are tossed out.

But with most trick dice, wouldn’t the independence assumption still hold? Then five-of-a-kind wouldn’t be any more likely on the next roll than it would be on any of the other subsequent rolls.

Depends on how the dice have been adulterated, I would think. If you suspect the equipment, you shouldn’t make any assumptions either way.