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’s the same way tossing a coin for the 50th time in a row still has 50/50 chances of heads and tails, regardless of what outcomes the previous 49 tosses had. The statistics terms is that these attempts, at coin tossing or playing slots, are independent events. Yes, it becomes more unlikely to keep getting some specific sequence of (non-winning) outcomes, but that’s only true when you step back and do the math for the whole sequence. It doesn’t change your chances of winning on the very next attempt.
You’re describing the gamblers fallacy.
Think of flipping a coin. If you flip a coin 50 times and it comes up heads every single time. On the 51st flip there’s a still a 50% chance of heads and a 50% chance it’s tails. The previous flips are not relevant to those odds.
For your slot machine, each spin has a 1/100 chance to hit the jackpot, that’s not going to change no matter how many times you play. Every play has the same odds of hitting the jackpot irrelevant of the number of times you play or the results of previous plays
Each individual spin has the same chances of winning: 1/100. This chance of winning is not dependent on the previous result, or the next result. The odds of hitting the jackpot are the same whether the jackpot has not been hit in 100 attempts, or if the jackpot was hit in the previous attempt. Each attempt will always have a 1/100 chance of winning
Because the trials are independent.
What’s the probability of flipping a coin tails twice in a row (assuming you only flip twice). Well, it’s 0.25. There’s a 0.5 chance of the first flip coming up tails, and a 0.5 chance of the second coin coming up tails.
But suppose you’ve *already* flipped tails once. What’s the probability the second flip comes to tails? Well it’s a coin, so there’s a 0.5 chance. The first flip is irrelevant. The coin doesn’t *remember* the result of the prior flip.
Thus we say that the flips are independent. The result of one flip has no bearing on the result of the next. The odds of getting two tails are 0.25, but only before the first flip. *After* the first flip comes up tails, we already have one guaranteed tail. All that’s left is the second flip, and like all coin flips, there’s going to be an equal chance of getting either possible result.
So if you’re in a casino right when it opens, you can look at a slot machine and say *the odds of it not paying out at least once in the next hour are a billion to one”, and be absolutely right. But, if you’ve been playing for 58 minutes with no wins, that has no bearing on what happens in the next two minutes. The incredibly unlikely event of not winning anything in 58 minutes has come to pass, but that doesn’t change how the machine works. It’s the past. And the next time you pull the lever, the odds of that *individual pull* winning is exactly the same as it was for all the other pulls.
To tie it into your math directly: the chance of it not hitting the jackpot on your first spin is .99^1, as you say, but it’s the chance of you not hitting the jackpot *on your first or second spin* is .99^2. And the .99^n is the chance of not hitting the jackpot *on all n spins put together*. But, if you’ve missed the jackpot 50 times, you aren’t concentrating all the odds of hitting the jackpot in that 51st spin. Rather, you already know the outcomes of those first 50 spins, and they don’t factor into the odds of the next spin, which remains .99.
What you said is mostly true. Slot machines won’t pay out jackpots more often than a pre-programmed setpoint, but once that limit is reached the probability doesn’t increase over time. It basically just “unlocks” the jackpot.
Anyway, for any game of chance with a set probability of success, previous results **do not matter**. If you’ve played 1000 games and lost 1000, you have the same probability of winning after that point as someone who just walked in the door. You aren’t “owed” a win or anything like that. Now the overall probability of going that dry may be low, but that has no bearing on future results.
The longer you gamble, the more likely you are to lose. In every game (excluding skill) the house is more likely to win than you are.
Aside: look up sunk cost fallacy, conditional probability, and binomial distributions
Because losing doesn’t increase your chance of winning the next roll.
If you roll a 10 sided die 9 times and never roll a 10, the chances of rolling a 10 on the 10th roll are still 10%.
Probability can be a bit confusing.
If you were to ask “If I roll the dice X amount of times, what are the chances I land on 6 one of those times,” your odds of getting a 6 would go up with the number of rolls. This is because you’re asking *what are my odds overall.* This stays consistent as we roll dice, as long as the probability we’re asking about continues to be “if I roll X more rolls, what are my odds of getting a 6.”
However, something being “due,” is **asking a different question**. It’s asking “what are my odds on the *next* roll,” which is completely different. Your odds for the *next* roll will always be 1/6. There’s nothing about your previous roll that can somehow instruct your next roll to be different, so it will always be 1/6.
The issue is independence vs. dependence.
A coin flip is independent. Prior coin flips have no impact on future ones. You cannot be ‘due’ to win a coin flip because there is no memory in the system – the coin doesn’t remember how it landed in the past.
A deck of cards is dependent. If I draw the Ace of Spades, I know that the remaining draws will never contain an Ace of Spades. If our game is won once the Ace of Spades is drawn, the fact that we have not yet drawn it increases the odds of drawing it the deeper we get into the deck.
A large part of the reason people tend to confuse these is the law of large numbers. This states that if you have enough trials, the average will be the odds. People have a great deal of trouble understanding how the coin ‘knows’ to regress to the mean if it doesn’t have memory.
So if I flip a coin and it comes up heads 5 times in a row, people assume that it’s more likely it will come up tails to ‘balance out’ the odds. But this isn’t the case. The reason it isn’t the case is that the coin doesn’t need to ‘know’ anything. Those 5 heads are being ‘washed out’ in the average by subsequent trials – if you flip the coin 1000 times afterwards, the fact that you had 5 heads at the start is trivial even though it seemed significant when it occurred.
you’re semi-right, which may be contributing to your confusion. the probability of getting a losing streak that long is exactly what you calculated. however the wheel spin doesn’t care or remember what any of the previous spins were. so it could be spin 1 or spin 10,000 – the probability of that particular spin is still 1/100
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