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
Because you’re stipulating (and forgetting) that something improbable has already happened, with a probability of 1.
After ANY 5 random flips, the chance that you’ll complete a run of 6 tails is
* (the chance P that you’ve already flipped 5 tails) x (the chance that your next flip will be a tail). [A]
Read that bit again until you’re happy with it, because it’s the key.
Before you started, the chance that you’d flip 5 tails in a row was 1/32. The chance that you’d then flip another was, as always, 1/2. So the chance that you’d flip 6 was, as you say, 1/64.
BUT. Your flips HAVE NOW HAPPENED. We have more information. You either flipped 5 tails or you didn’t, and we know which it is. The chance P that you’ve already flipped 5 tails isn’t 1/32 – it’s either 0 or 1. The chance of your completing a run of 6 flips is either 0 x 1/2 = 0, or 1 x 1/2 = 1/2.
You’ve stipulated that you’ve already flipped 5 tails. The probability of that isn’t 1/32 any more – it’s 1. The probability of a 6th tail is 1/2.
(Try *reductio ad absurdum*. You could use the same sort of logic as you feel you want to for runs of 6 flips, for shorter runs. So how about runs of 2? The chance of two tails is 1/4, so if you’ve just had a tail, it *ought* to feel (to you at least) like a head is more likely. Similary the chance of two heads is 1/4, if you’ve just had a head, it *ought* to feel like a tail is more likely. So do it. Start flipping a coin twice, repeatedly, and record happens. It shouldn’t take you more than a few dozen trials to convince yourself that what the first flip was makes no difference.)
Because whatever happened to the 5 coins won’t change what happens to the next one. Definition of independent is they don’t have an effect on each other. Whether you flipped 5 heads or tails or anything in between, it doesn’t matter because when you flip the next coin, you flip it in isolation. Independent. Alone. Don’t look at it as flipping 6 coins.
As everyone has said, the previous flips don’t matter. For me it’s easier to think of it this way though; the possibility of flipping TTTTT is the exact same as TTTTH or HHHHH or HTHHT or TTTHT. Every possible 5 combination has the exact same odds, but you’re only going to get 1 of those combinations. No one combo has any higher possibility than another, every combo has a 3.125% of occurring.
The odds of flipping one tails is 1/2. The odds of flipping 5 tails is 1/32.
If you’ve *already* flipped 4 tails though, you’ve already satisfied 1/16.
What’s the difference between 1/16 and 1/32?
1/2, the odds of flipping one tails
Imagine you had 6 different people flip a coin at the same time, and they wrote down the result. You read the results one at a time. After you read the first 5 results and they’re all heads, do you feel the 6th one should be tails? Would it feel wrong if you read it and it said heads as well? What if I told you that all of the people flipped their coins on different continents? Do you feel like someone flipping a coin in Paris would change your chances?
What I am trying to show is that there is no link between the flips, no way they could influence each other. It’s the same even if it’s the same person flipping the same coin, there is no link between flips, not in any way that has any effect on a flip.
The reason you “feel” it should is because humans evolved to notice patterns. Specifically in this case you are noticing the flips are not making the pattern you expect, so you feel that it should overcorrect to make it look like your expected pattern, like it’s “due”, so that the pattern will be more like you expect.
The key is to realise that the sequence “TTTTTH” has the same probability of occurring as “TTTTTT”, and so does “THTHTH”, “TTTHHH”, “TTHHTT” and indeed any set sequence of 6 flips, which are all 1/64. Just because you’re “looking for” or placing some special significance on 6 tails in a row doesn’t actually make that sequence special at all.
In other words you can’t think “oh wow 6 tails in a row is really unlikely”, because “5 tails and a heads” is just as unlikely (1/64), it just doesn’t *seem* as “special”.
So when you’ve rolled 5 tails already, that result had a 1/32 chance. When you flip the next coin it’s a 50/50 which halves the chance of the total sequence to 1/64, then the next roll halves the chance of that total sequence again (or you could view it as doubling the number of possibilities for that number of flips) to 1/128, and so on.
You could flip TTHTHH, you could look at it and say “that’s just a random jumble, meh”, and then flip another 6 times and get TTTTTT and say “oh wow! That’s amazing! How unlikely!”… and if you wanted to see TTTTTT again you’d have to roll a set of 6 another 64 times to see it. But if you were to set out to get TTHTHH again, you’d have to try an average of 64 times to get it again, too.
You’re conflating independence and probability of consecutive events.
Every coin flip is independent and has a 50/50 chance.
However, there is a 31/32 chance that you will flip tails at least once over the course of 5 flips.
It doesn’t explain when that flip will turn up though.
Each flip is independent. The universe does not conspire to even out results over multiple flips. Think about how would that even work? There’d have to be some memory stored of what the previous results were and some mechanism to alter physics to influence the probability of the next flip. Seems unlikely to me unless you believe there is something non-scientific outside the laws of physics happening like fate
Here’s where I will always disagree with the “standard” answer, at least as far as a larger number of flips (say 100) goes. If I flip a coin 99 times and it lands on tails 99 times in a row, the odds against that happening are astronomical! This would lead me to conclude there is another factor at play. I would question whether the coin is weighted differently, or if there is some sort of strange magnetism or environmental condition happening, or even a strange gypsy curse on the coin, it doesn’t matter. It would be so phenomenal to have 99 flips land on tails, it is actually **more** reasonable to assume that whatever phenomenon has caused this will also cause it to land on tails again.
Consecutive sequencing. Isolated it’s a 50/50 chance but when you have already flipped 5 or 6 tails in a row, you’ve already experienced the unlikely occuring so the probability of the next one being tails lessens.
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