Scenario: there is a machine at a casino that hits jackpot 1/100 times it is used. The probability that one does NOT hit jackpot on their first spin is .99^1, the second .99^2, and on their nth .99^n (hoping my math is right). As the number of non-winning spins increases, many people would say the machine is “due” because the probability of the losing streak continuing gets lower and lower, but AFAIK that is not valid. Why is that?
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Sometimes it is, and sometimes it isn’t. It depends upon the gamble.
Look at it this way.
You put 10 stones into a bag, one of them blue, the others white.
If you pull out a stone and it isn’t blue, set it aside, and keep going, then eventually you will get the blue stone. Eventually, you will run out of white stones.
However, if you get the blue stone and set it aside, your chance of getting the blue stone on the next draw is zero.
The odds of you getting the blue stone are dependent upon what stones you previously drew and how many.
However, if you put the stone back every time you draw, then the later odds are independent of the earlier ones. If you put back the white stone, then there are always 9 white and one blue, and the odds are the same. Similarly, the odds of getting the blue stone do not change if you got the blue stone last time if you put it back. Still one chance in ten.
In the first case, it matters how many stones have been taken out and not returned, and what color they are. In the second case, it is always the same as the first draw: there are always 9 white and one blue, and your chances of getting the one blue are always 1-in-10.
The hard part is figuring out whether or not the number of previous tries changes the odds, i.e. the odds are dependent or independent.
Counting cards is an example of correctly seeing that the odds of each card in the next deal is dependent upon what is played. The next card cannot be an ace if all aces in the deck have been played, for example. Another example is pull tabs. If there are five $50 pull tabs in a stack of one thousand, no one has gotten one, and there are only 5 left, you cannot fail.
A lottery where the ball is drawn from a properly functioning randomizer or spinning on a properly functioning roulette wheel, not so much. Hitting the number 23 once doesn’t remove 23 from play, nor is there anything forcing the wheel to hit one number more than the others.
In some cases, it is a bit of both. For example, slot machines are designed to hit more the longer they are played, or at least used to be. This was intended to get people thinking that if they just kept playing they would come out ahead. They were, of course, designed to do this so little that it almost never was better to just sit there and play. However, watching a slot machine fail to hit over and over until other people had given up meant your odds on that machine improved.
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