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

Coin flips are independent events. The values of previous coin flips do not affect the values of future coin flips. Believing otherwise is known as the Gambler’s Fallacy.

Some events are not independent or are otherwise cyclical. In that case, knowing they have not happened for a long time can tell us something about the probability that they will happen now. Say I select my socks every morning at random and have one pair of purple socks. Each morning I select a non-purple pair of socks, the probability of selecting purple socks the next morning goes up (conversely, if I pick purple socks this morning, my chance of picking them next morning craters to 0). This doesn’t violate the Gambler’s Fallacy, because my sock selections are not independent – if I wear a pair one day, I can’t wear them some other day.

So the trick is to think about whether the events are independent or not. Most of the things we do in games of chance (flip coins, draw cards, generate random numbers, etc.) are independent. Many of the things we deal with in real life are not. The Gambler’s Fallacy mostly arises from us taking our real-world instincts into a special place (like a casino) where they don’t apply.