question about odds, probability and statistics

[dehro]

this is something that has been bugging me for as long as I've spent time thinking about it.. and I'm not even sure I know how to put it in the right words.

when you roll a d20 you have a 1 in 20 chance to roll a certain number.
same with a raffle ticket, you have one chance in however many tickets have been sold.
why is it then that I read people claiming that it's statistically unlikely for a number not to turn up the more you try?.. how can someone say that if you roll that dice a million times you get anything different than 1 in 20 chances to roll a number and, conversely, of never rolling it?
the fact that a certain number hasn't turned up for say, half a million rolls doesn't make it any more likely that it will turn up in the next roll, doesn't it?
am I thinking wrong?
(I'm of course talking about the abstract situation..let's not introduce loaded dice in the equation :P)

[Lorsa]

The chance that you will roll a 1 on a d20 is always 1/20 (disregarding that a die is actually based on Newtonian physics and is actually not a true randomizer). Regardless how many 1's you rolled before (or not), the chance is always !/20. Any single outcome is not dependant on what came before.

However, any single sequence of numbers also have a probability linked to it. So rolling ten 1's in a row has a probability of (1/20)^10. But the probability of any single roll is still 1/20. So no, a die roll is not affected of what came before. Neither is a coin flip etc.

How raffle lotteries work I don't know. I thought the probability depended on how many lottery tickets were printed rather than how many were sold. If it depends on how many were sold then obviously the chance you will win with your single number decreases the more tickets that are bought.

[Eldan]

You are quite right here. The chance that the next die roll will be a 20 is always 1 in 20.

However, if you roll a million times, the chances are very high that at least one of them will be a 20. (The chances of not rolling a twenty is (19/20)^(10^6), or so small that the google calculator rounds it to 0).

[Teddy]

Just a question, do the people who say this do so during the rolling sequence or before it, because if they have some idea of cheating destiny (for lack of a better word), they're doing it wrong, but if they're reasoning that if I roll 100 d20s, then I should have a greater likelyhood of rolling at least one 20 than if I roll just one d20, then they are indeed correct. Now, given how many gamers are prone to gamer superstition, I assume it's the former, right?

Now, regarding gamer superstition, well, I can't speak for anyone but myself, but whenever I indulge in it, it's more in a way of honing tradition than me actually being serious about it, and I suspect this might be quite widespread. So don't start calling out people on their poor use of statistics unless you're sure they're actual kook and not just doing it for fun.

[Killer Angel]

Mandatory link (http://www.darthsanddroids.net/episodes/0099.html).

[Lorsa]

However, if you roll a million times, the chances are very high that at least one of them will be a 20. (The chances of not rolling a twenty is (19/20)^(10^6), or so small that the google calculator rounds it to 0).

Matlab doesn't seem to like it either. It does say that it's a 1.7221 * 10^(-223) chance of not rolling a 1 if you roll ten thousand times. Another power of ten rolls and it gives 0.


However, what would be the odds if you add in possible mind-control of the dice or the local probability laws? Is that something you can figure out with pseudoscience?

[TuggyNE]

However, what would be the odds if you add in possible mind-control of the dice or the local probability laws? Is that something you can figure out with pseudoscience?

You can figure anything out with pseudoscience.

[Asta Kask]

why is it then that I read people claiming that it's statistically unlikely for a number not to turn up the more you try?.. how can someone say that if you roll that dice a million times you get anything different than 1 in 20 chances to roll a number and, conversely, of never rolling it?
the fact that a certain number hasn't turned up for say, half a million rolls doesn't make it any more likely that it will turn up in the next roll, doesn't it?
am I thinking wrong?

Humans are not very good at probabilities like this, where the chances are independent. We've evolved in an environment where chances are dependent, and then different rules apply. Casino's use roulette wheels and dice precisely because they are counterintuitive - their money is made in the gap of our knowledge.

[factotum]

Humans are not very good at probabilities like this, where the chances are independent.

Humans are also extremely good at pattern-matching, to the extent that they will see patterns where none exist. Someone may well be convinced that a particular dice is really good for them because they "always roll 20s" on it, but the chances are that the dice is just as random as any other d20 and they're just imposing a pattern on the randomness that isn't actually there (maybe they got three 20s in a row one time and have been holding this odd belief ever since).

[Socratov]

well, even with a pseudo scientific method you can make more predictions:

indivdually every roll counts on itself with no history applying, however, in a series of roll you could say that if you rolled a 1 first that there after another 1 is a slim chance.

when having rolled a 1 the following applies:

a 1 has been rolled, so the chance of another 1 being rolled in the series is slightly less, assuming the chance of rolling a number are equal for all the numbers.

this would suggest that rolls are affected by their rolling history even though each roll is a fresh 1/20 chance. In a series however, you can, via assumption of equal statistical distribution, assume some numbers have higher odds of rolling then others.

It seems vague, but if you woudl roll a couple of times in series and monitor the results you will see a significant difference between pairs in set and not paris in set.

[Lorsa]

this would suggest that rolls are affected by their rolling history even though each roll is a fresh 1/20 chance. In a series however, you can, via assumption of equal statistical distribution, assume some numbers have higher odds of rolling then others.

It seems vague, but if you woudl roll a couple of times in series and monitor the results you will see a significant difference between pairs in set and not paris in set.

I think there game "Lotto" is being played not only in Sweden but also in other countries. If what you say is true then you could create a mathematical formula based on what numbers had previously been generated to predict what numbers would come up next. That no one I know is doing this (and any statistical mathematics teacher would advice you against playing Lotto altogether) is, if not proof that it doesn't work, at least a strong indicative that you can't predict numbers based on previous numbers.

Of course there's a difference between pairs and not pairs. A pair coming up in series is basically 1/20 for a d20 die whereas a non-pair coming up is 19/20. It's because if you rolled a 1 there's 19/20 chance that 1 is followed by any other number thus not giving a pair. Which is, strangely enough, the same chance to roll a one vs a not-one in the first place.

There is no possible way to predict what the next result of die roll would be (except by using Newtonian mechanics as stated before) using any arbitrary length of rolling history. The result will always be 1/20 chance for any given number.

[Socratov]

I think there game "Lotto" is being played not only in Sweden but also in other countries. If what you say is true then you could create a mathematical formula based on what numbers had previously been generated to predict what numbers would come up next. That no one I know is doing this (and any statistical mathematics teacher would advice you against playing Lotto altogether) is, if not proof that it doesn't work, at least a strong indicative that you can't predict numbers based on previous numbers.

Of course there's a difference between pairs and not pairs. A pair coming up in series is basically 1/20 for a d20 die whereas a non-pair coming up is 19/20. It's because if you rolled a 1 there's 19/20 chance that 1 is followed by any other number thus not giving a pair. Which is, strangely enough, the same chance to roll a one vs a not-one in the first place.

There is no possible way to predict what the next result of die roll would be (except by using Newtonian mechanics as stated before) using any arbitrary length of rolling history. The result will always be 1/20 chance for any given number.

Well over the course of enough datapoints you could generate a distribution of numbers who are above and below par (copmpared to a theoretical distribution), increasing the likelyhood of guessing the right numbers

[Darth Credence]

Well over the course of enough datapoints you could generate a distribution of numbers who are above and below par (copmpared to a theoretical distribution), increasing the likelyhood of guessing the right numbers

But you would only increase the likelihood of guessing the correct numbers if the die was unfair. While the more often you roll a die, the closer it will come to all values being equal, it would be a huge statistical fluke if they ever were exactly equal. If you have rolled a d20 twenty times, and somehow gotten each number 1-19 once, the odds that you would end up closing that out with a 20 to make everything even is only 1/20. In any large set of dice rolls, some numbers will come up more often than others, simply because it is random and the dice have no memory. But, unless the die itself is skewed in some way, you cannot predict future results from those rolls.