Flipping a tail is a 1/2 chance, but flipping 6 tails in a row is a 1/64, so if after flipping 5 tails, why is it incorrect to say that your chance of flipping another tail is now lower, like you’re “bound” to get a head? I know this is the gambler’s fallacy, but why is it a fallacy? I get that each coin flip is independent, but it feels right (as fallacies often do) that in consecutive flips the previous events matter? Please, help me see it in a different way.
In: Mathematics
Streaks are normal in a something that is random. If you want to see a fun example of this, check this [numberphile video](https://www.youtube.com/watch?v=tP-Ipsat90c)
Simple answer: flipping TTTTTT has a 1/64 chance, but so does flipping TTTTTH. They have the exact same probability. So when you already got TTTTT those are the only two options left and they have the same probability.
The odds of flipping XXX XXX or OOO OOO…
… are exactly the same as you flipping XXX XXO or XOX OXO.
Each individual flip is 50-50, but the odds of achieving ANY specific pattern of 6 is 1/64.
>I know this is the gambler’s fallacy, but why is it a fallacy?
Because each flip is a discrete event, and the prior events have no impact on the future events. There’s no law or mechanism of the universe that’ll reach out and say “okay, that’s enough tails, now you get heads.”
The coin doesn’t care what happened before. Each time you flip it, there are only two options: head or tails.
Consider the following 6 flip sequences:
h-h-t-t-h-h
t-h-h-h-t-h
h-h-h-t-t-t
h-t-h-t-h-t
t-t-t-t-t-h
t-t-t-t-t-t
Every single one of them has the exact same probability of occurring, 1/(2^6) = 1/64. Every single coin flip sequence of n flips has a 1/(2^n) probability of occurring if you flip a coin n times.
If you want to, you can take the time to write out all 64 possible unique sequences of 6 coin flips. Since there are 64 sequences and one of them is t-t-t-t-t-h, this sequence has a 1/64 chance. Same for t-t-t-t-t-t.
Every time you flip a coin, your chances of getting tails is 50%. What happened on previous flips doesn’t impact what will happen on this flip. Each flip is a unique event unrelated to anything that has happened before or will happen in the future.
Lot’s of good answers, but I’ll try to explain it more like you’re 5.
Every single time you flip a coin write the result on a piece of paper, and imagine everybody in the world is doing the exact some thing, and if they don’t get the same result as you the stop. After one flip, half of the world just stopped. After two flips, half of the people still flipping stop. After three flips, another half, and so on. This is true no matter what you get on each flip. If you wrote heads on that paper five times and you flipped tails, half of people flipping would stop because they got heads. Those people just flipped heads six times in a row and on the last flip their odds were 50/50
Or
If i hand you a fair coin, the odds of getting heads or tails is 50/50. No matter how many times i flipped it in the other room and what the coin landed on each time it’s still a new coin for you and you have 50/50 odds.
Each flip you make no matter what you get, the next flip is still just taking a new coin and flipping that new coin which has 50/50 chance of landing heads or tails.
The odds of one flip never change, only trying to predict more than 1 flip ahead of time changes how likely that is to be right.
Let’s say you flip five tails in a row and then drop the coin on the ground and lose it. A week later, somebody finds the coin and flips it. What are their odds of getting tails? 50/50, right? Why? Because the coin doesn’t store luck. It doesn’t store luck over the course of a week and a change of ownership and it doesn’t store luck over the couple of seconds that it takes to flip it again. They’re independent events.
Lets just imagine that all previous flips matter. Wouldn’t that mean that you would have to know ALL its previous flips to be able to determine if its “bound” to flip heads?
Eg. You pick up the coin and it flips 5 tails, but if before that someone else had flipped 20 heads its still “bound” to flip tails. Even if you knew all the flips a coin has made it wouldnt really make a difference that you had flipped 5 tails since its probably been flipped thousands of times 5 flips in a row is nothing in a ever expanding dataset.
But since all the other flips already happened it doesnt really influence the next flip. If we exaggerate the example its easier to see.
Lets say you and I make a bet, I say: “Ok, If I win the lottery and lightning strikes me and I roll at least one head in ten coin flips you give me 50 bucks otherwise I give you 50 bucks”
Sounds like a great gamble for you right? So you accept.
Then I say “Ok well it turns out I already won a small lottery 15 years ago and I was hit by lightning 20 years ago”, so now the gamble is terrible and logically you lose as I dont flip ten tails in a row.
But nothing changed really, why did the gamble go for good to bad after revealing that something already happened? Well, because if it has already happened you aren’t working with the probability of that happening or not since its 100% happened so you only work with what has not happened yet.
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