cjo20 t1_ja9nht3 wrote on February 27, 2023 at 9:27 PM

Reply to comment by johrnjohrn in ELI5: why does/doesn’t probability increase when done multiple times? by Reason-Local

The “absurdly unlikely” part comes in to play in being able to view the events from two different perspectives.

One is that, if someone claimed at the outset, that they could predict the next quintillion rolls of the die (whatever values those might be), the probability of all of them being correct is vanishingly small - each of the 6^quintillion combinations almost certainly won’t show up, only one will, and you’re relying on picking that one sequence.

However, once you’ve correctly predicted the quintillion rolls in a row, if you then say “I’m going to roll a 6 next”, you aren’t any more or less likely to get it right than you were on the first roll.

The probability of being able to predict (N+1) correct dice rolls is N * 1/6.

1 roll: 1/6

2 rolls: 1/6 * 1/6 = 1/36

3 rolls: 1/36 * 1/6 = 1/216 Etc.

If you’ve already done the N dice rolls, you’ve already dealt with the probability of getting to where you are in the chain of rolls. The probability to advance to the next step in the chain is always the same though, even if the chances of you successfully getting to that point in the chain are infinitesimal. You’d still expect 5/6 to get it wrong at the next roll, 35/36 to get it wrong in the next 2 rolls, and 215/216 to get it wrong in the next 3 rolls.

johrnjohrn t1_ja9oihv wrote on February 27, 2023 at 9:33 PM

You have done a good job of explaining the math. And thank you.

Now, you're sitting at that gambling table and someone gives you the opportunity to choose one number that will appear on the die for the next roll after one quintillion. Are you going to choose some number other than the one that came up one quintillion times or some other number? I imagine you would choose the same number instead of picking some other number at random.

cjo20 t1_ja9p3u1 wrote on February 27, 2023 at 9:37 PM

Ultimately, it doesn’t matter which number I pick, I’ve still got the same chance of being right as with any other number.

People can use superstition to help them decide, but it doesn’t make them any more likely to be right. Some people will choose 6 because “that’s got to be right, it’s happened so often”. Others will choose their favourite number “because that’s lucky”. Others will choose anything but 6 because “they can’t be that lucky”. Any logic you try and apply to it to say “this outcome is more likely than any other” is just your brain tricking you.

johrnjohrn t1_ja9s99d wrote on February 27, 2023 at 9:57 PM

Would you not sit up at a story on the news where someone rolled 7's at a craps table for a year straight, only stopping to eat and sleep? If you paid any attention to that story would you be a fool? They bring officials in and claim the game is still fair and allow it to continue. Are you a fool if you claim it is rigged? Now that same roller rolls for multiple decades. Do you still calmly say, "we are foolish to assume this person will roll 7's one more time just because of the past 50 years they have continued to roll 7's. Each roll is a new roll." ?

cjo20 t1_ja9trp6 wrote on February 27, 2023 at 10:07 PM

If you’re trying to construct an actual scenario, a casino wouldn’t let that happen. They’d kick the player out because “they believe them to be an advantaged player”, because they don’t like losing money. And eventually you reach a point where it’s simply more likely that there is a bias somewhere in the system that hasn’t been detected yet.

That means it would be a feature of the system (player / table / dice) rather than of the maths - the maths is based on perfectly controlled probabilities.

Practically, you can’t ensure it’s a 100% fair system, so the simple “each outcome is 1/6” breaks down. If you could guarantee that it was perfectly fair, then what I said earlier stands. In a Real-World situation, the assumptions change significantly - you can’t have perfect knowledge of everyone’s intentions, whether it could be a scam etc.

EDIT: however, most gamblers fallacies aren’t based on the idea “I have actual evidence that the system is rigged”. Things like “5 hasn’t come up on the roulette wheel, it must be overdue” aren’t based on an assumption of bias, they’re based on an assumption of fairness, which says that eventually all numbers will come up equally. However, they don’t have to come up equally before the heat death of the universe.

johrnjohrn t1_ja9vmsv wrote on February 27, 2023 at 10:19 PM

I'm not trying to construct an actual scenario. I am constructing a hypothetical scenario that says there is no chance that the system is rigged, and there are a quintillion throws that are all identical, which is entirely possible, but highly improbable. In real life we can say, "that would never happen", but the math says you are incorrect and it 100% could happen. Now, this situation, which is mathematically possible, plays out (hypothetically). Which bet are you going to make after the one quintillionth throw? And are you a fool if you use past information to say the next throw will remain the same as the past quintillion?

cjo20 t1_ja9x1ev wrote on February 27, 2023 at 10:29 PM

Again, if it’s guaranteed to be mathematically exactly fair, then by the maths I posted earlier, claiming you have better than 1/6 chance of getting the next one right is mathematically impossible, by definition.

To be clear: you’re defining a situation whereby you are guaranteed to only have a 1/6 chance of getting the next number correct, whichever you pick, and then saying “isn’t it better to stick with the number that came up before?”. Simply, no, it’s not, because of the way you defined the system.

Monimonika18 t1_jadhi4i wrote on February 28, 2023 at 5:29 PM

Thanks for pointing out that commenter's moving goalposts.