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

Before you start flipping the coin, flipping 6 tails in a row is indeed pretty unlikely. However, by the time you’ve flipped it 5 times, the “unlikely” part has already happened. Taken as a whole, the sequence is unlikely, but the fallacy comes from the fact that at the 5th flip you’re not dealing with the whole sequence anymore, but just that single 50-50.

Think of it another way: if I showed you two (perfectly ordinary) coins and told you that one of them just flipped 6 heads in a row, what could you possibly do to determine which one of them is more likely to land tails on the next flip? None of the flips you do change the coin in any way to make one outcome more or less likely.

You said it yourself. Each coin flip is independent. The coin doesn’t know about what happened before, it will always give you a 50/50 chance for either heads or tails.

Think about it this way. If you were to toss a coin five times and you get heads every time, then you could give that coin to another person and ask them to toss it for you. It will still be a fair 50/50 for heads or tails. They don’t know the previous result, why should the probability be any different for them?

independent random events every flip has the same 50/50 odds. this is the same concept slot machines use

with a large enough sample size it will always end up with 50 50 but in a smaller sample size that may not show up

you mention flipping 6 in row and flipping once the two are completely separate from each other

Let’s simplify this a bit and flip 3 coins in a row instead of 6 coins in a row.

There are 8 different possible outcomes (H for heads, T for tails):

HHH HHT HTH HTT THH THT TTH TTT

So let’s say I flipped 2 tails in a row. Which means that after my next coin flip, the results are going to be either TTH or TTT. Either one of these sequences has a 1 in 8 chance of happening when flipping a coin 3 times in a row. In other words, you are exactly as likely to flip 2 tails in a row followed by a heads, as you are 3 tails in a row. That’s why the last coin flip is still a 50/50 chance.

So going back to your example, the chance of flipping 6 tails in a row is 1/64, but the chance of flipping 5 tails in a row followed by a heads is also 1/64.

Provided it’s a fair coin, each flip is an independent event with a probability of 50% heads, 50% tails, you are correct. The coin has no memory; it doesn’t care what happened in the previous flips.

This does mean that over a large number of trials (flips) you would expect approximately the same number of heads and tails.

This does create something of a logical paradox because, if heads is “winning” 5-0 and over time you expect equal numbers, the probability of tails appears to have to increase.

It doesn’t; it remains 50/50

>it feels right (as fallacies often do) that in consecutive flips the previous events matter?

Why does it feel right?

How would a coin know what was the previous flip?

What would the mechanism be?

Do you believe in some force like luck or fate or God controlling what happens with the next coin flip?

The math behind what you’re referring to is called conditional probability. So the probability “A” (that you flip 6 tails in a row) given that “B” (you already flipped 5 tails in a row) is Prob(A) / Prob(B) = 1/64 ÷ 1/32 = 1/2.

But conceptually, you just have to realize that flipping a coin is an individual event. Think of it this way… If you roll a 6 sided die, the probability of rolling an even number is also 1/2. So your odds of rolling an even number on a die and then flipping a tails on a coin is 1/4. Let’s say you roll a die and get an even number. So now that you’re about to flip the coin, do you think the roll of your die will affect the flip of your coin? Would your odds of a tails be different if you had rolled an odd number?

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