Just assume they are fair dice. In which case you should definitely bet on the next roll.

Las Vegas greatly increased the amount of people playing roulette and the size of the average bet when they installed lighted signs that show the result of the last dozen spins or so. People will look at a string of Black numbers in a row and assume that Red is now due.

The House laughs all the way to the bank.

Because, If you chose the next roll there would be 1/1,296 chances of it happening again. If you chose the second roll after, there would be 1,295/1,296 X 1/1,296 chances, for the third roll 1,295/1,296 X 1,295/1,296 X 1/1,296 chances and so on.

So if you had to pick a roll for the next occurrence, the best bet is - next one.

Ah, so you meant “bet on which subsequent roll you will *first* achieve the feat again.”

Yes. I like to point it out to people because once you understand it you are somewhat less amazed by coincidence.

What I shoulda said was

Does this mean that I can expect a four of a kind once every 52 or so times *I* roll the dice OR…

Okay, let’s assume 52 coffee shops running the same game, albeit with their own sets of dice. Each week I visit a different one and play. Can I expect to roll four of a kind about once a year? How are the odds distributed now that I’m playing with 52 different sets of dice? It does seem to me that I should still expect to roll four of a kind about once a year.

Yes.

It doesn’t matter whether you use the same dice at the same place, or different dice in different places. You should expect about 1 success in 52 trials.

Of course, you might get no wins, or you might win every time. But on average, you’ll win once every 52 trials.

I think where people get lost on this (and I have seen a friend do it in a casino) is conflating when and what bet is made.

If they see a fair die rolled 10 times and no 6 comes up they think that the chances of that happening are very low. If they were to bet that no 6 would come-up after 10 rolls they would think it is a long shot and each roll past that just makes the odds longer. So, surely, after 10 rolls with no 6 the 11th roll must be more likely to come up 6!

Nope. Of course not. Still a 1:6 chance.

My friend at the casino saw a tally of the last ten (or whatever, I forget) spins at a roulette table. There had been something like five blacks in a row so she insisted red was due. I tried to explain to her that is not the case but she insisted it was because the chances of five blacks in a row is low and six would be lower still thus red was due (and her day job was an office manager and she kept the financial books, you’d think she would have known better).

Note that this is where the concept of variance (or standard deviation, if you prefer) helps.

The bigger the variance, the more trials you’ll need to get to the ‘expected’ long term behavior.

On average, you may see 4 of a kind once a year (assuming fair dice and one try a week), but to actually get close to an average of once a year may take several years of once a week trials. Some people will hit multiple times a year. Some won’t hit at all for several years. To guarantee an average of ~1x a year would take several years or averaging across several people.

Think of it in terms of coin flips. Sure, you’d expect to see heads exactly once every two flips, but half the time you get either 0 or 2 heads instead. To guarantee something close to a 50/50 distribution (say 45-55% heads), you should flip several times in a row (~30 or so should get you there with reasonably high probability).

Incidentally, the 1/52 is probably close enough to go on, but the true figure is 25/1296 (1 in 51.84)

What does this mean to you? Are you thinking of your odds over the current calendar year or over the next 52 weeks starting every time you roll?

Before you start, the odds are that you will receive one 4 of a kind if you roll every week for 52 weeks. After the first week (Jan 1), assuming you don’t win, the odds remain the same - you will receive one 4 of a kind if you roll every week for the next 52 weeks. Your odds of getting 4 of a kind **over the remaining 51 weeks of the calendar year** however are not as good. For every week you don’t get 4 of a kind, your odds of getting one before Dec 31 go down.