A question about probability

Question

I recently had a disagreement with a friend about what it means for an event to have a probability of happening every X times.
Take for example a coin that if flipped has a 50% probability of landing heads and a 50% probability of landing tails.
My understanding is that while the probability of landing tails 3 consecutive times is smaller than that of landing it 2 consecutive times, if in the last two flips you got tails it has no effect on the probability of you getting tails in the following coin flip.it stays 50-50.

But he seems to believe that the probability of you getting tails gets smaller and smaller the more you land multiple consecutive tails.which sounds very counter intuitive to me.

For a more concrete example, he seems to think that if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high while I think it had the same probability of occuring today than it had of occuring 9 years ago.

Which one of us is wrong, please give examples.

In: 5

Answer 1

You’re exactly right. Your friend has fallen victim to something called the [gambler’s fallacy](https://en.wikipedia.org/wiki/Gambler%27s_fallacy).

So while it is very unlikely, for instance, for a coin to land heads 100 times in a row, it does not make it more or less likely to land heads on the 101st flip.

>For a more concrete example, he seems to think that if an event has a probability of occuring every 10 years, and it’s been more than 10 years since the last time it occured, we are living on borrowed time and the probability of it happening now is extremely high while I think it had the same probability of occuring today than it had of occuring 9 years ago.

This is harder to answer because “probability of occurring every 10 years” isn’t well-defined. It could mean a lot of things. But I think you’re talking about a situation like this: lets say that there’s a volcano, and scientists estimate that every year it has a 10% chance of erupting. A naive interpretation might say that it will go off every ten years. But in fact, if we do the calculation we’ll find that there’s roughly 35% chance that ten years will pass with zero eruptions. If that happens, you’re correct: the probability has not changed, and there is no “borrowed time.”