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I would not be able to put this into symbols, but I ask here because I think it's the correct place to ask.

Would the chance of my parked car getting damaged (bumped or scraped) by other cars parking nearby increase over time?

Gambler's fallacy says: if something happens less frequently than normal during some period, it will happen more frequently in the future. (wikipedia)

For every time I leave my car parked, there is a chance it will get damaged. If I don't want to commit the gambler's fallacy, I should consider the chance of damage the same every time I park.

But if I park at the same spot every day for many years, the chance that my car would have been damaged after all those years, would surely be greater than if I just parked there one day, right? How does this not contradict the gambler's fallacy?

My insurance company asks a higher premium if I park on the street all year round, than in a garage, so somehow they must figure that the chance is higher than if I just park on the street one day. How does this not contradict the gambler's fallacy?

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The probability of your car being damaged on any given day is not increased just because it was or was not damaged on a previous day. But over a period of time, as time grows large, the probability of your car being damaged approaches 1.

The gambler's fallacy is when one assumes that an independent event has a higher probability due to it not having occurred previously. In this case the gambler's fallacy is not being committed, as what you're concerned with is the probability of your car being damaged over a period of time.

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If you for example let the probability of the car being damaged be $0.01$ on any given day, independently of what happened in the past, then the probability that you will avoid damage over the next year is $0.99^{365}\approx 0.025$. So obviously, the longer the period, the higher probability of getting damage.

Now, let it be the case that you have had no damage for the last $10$ years. Using the independence of events, the probability of avoiding damage in the course of the next year is still $0.025$. Committing gambler's fallacy here would be to say that this probability is lower than $0.025$ because of the long stretch with no damage. On the other hand saying that the probability will decrease if you take a longer time period, is not gambler's fallacy.

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