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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It depends on how the machine works but typically the probabilities are independent, meaning every time you play the machine the odds of winning are the same. In which case, as others have mentioned, this is an instance of the “Gambler’s fallacy”.
What *does* happen is that every time you play, some of the money you put into the machine is added to the jackpot. So there is in fact a benefit to playing a machine that hasn’t had a payout for a long time, not because you’ll have a better chance of winning, but because when you do win, your payout will be larger.
For instance, say we take your example of a machine with a 1-in-100 jackpot probability. On average, you have to play 100 turns on the machine to get a single jackpot, but much longer losing streaks are possible. Now let’s say someone who used the machine before you played for 200 turns without success. They put a quarter in the machine every time and 20 cts of each quarter was added to the jackpot, so the jackpot is now at $40. This means you can now afford to play 800 turns, and as long as you win the jackpot somewhere in those 800 turns, you make a profit or at least break even.
(You can afford 800 turns because as long as you win, each turn actually has a net cost of 5 cts and you start with $40 “in the bank”. $40/0.05 = 800.)
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