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
Six tails in a row is 1/64. But five tails followed by a heads is also 1/64. By the time you get your five tails, the chance of both next possible outcome are (and always have been) equal.
But the important thing is that 1/64 is only the initial probability. The probability of each outcome changes as new information comes to light (it must, since many outcomes that were originally possible now have probability of 0).
If I told you regular smokers have a 5% chance of dying to lung cancer, that doesn’t mean you can go up to someone with terminal lung cancer who is on hospice and say “don’t worry, there’s a 95% chance you’ll beat this”. That 5% figure represents all smokers – you can’t apply to any given subset of smokers as you see fit.
You’ve flipped a coin. It was tails. That was a 50/50 chance.
You’ve flipped a coin. It was tails. That was a 50/50 chance.
You’ve flipped a coin. It was tails. That was a 50/50 chance.
You’ve flipped a coin. It was tails. That was a 50/50 chance.
You’ve flipped a coin. It was tails. That was a 50/50 chance.
You’re flipping a coin. It might be tails. That’d bea 50/50 chance.
Flipping 6 heads in a row doesn’t mean you’re due for 5 tails to balance it out. It means that those 6 heads will be diluted by subsequent flips which will be roughly 50/50.
The reason is because your brain notices unusual patterns better than other patterns, but mathematically they’re the same.
If you flip 6 coins and get HHHHHH, your brain notices that. You would also notice TTTTTT, and you might notice HTHTHT and HHT-HHT.
It’s easy to get a cognitive bias of, “If this coin is random, then after getting HHHHHH I’m bound to get a T soon.” Likewise, if you got HTHTHTHTHTHT, you might start to expect a double heads or double tails soon. And HHTHHTHHTHHTHHT might make you think you’ll get a THT or TT somewhere to break up that unusual pattern.
Mathematically, all of these strings are equally likely. When you flip a coin once, H and T are equally likely. When flipping twice, HH, HT, TH, and TT are all equally likely. Likewise, when flipping three times, HHH, HHT, HTH, HTT, THH, THT, TTH, TTT are all equally likely strings. The cognitive bias is your brain notices HHH as unusual but does not notice HHT as unusual, even though they have exactly the same chance of showing up.
The odds of getting HHHHHH is indeed 1/64. However, the chance of HHTTHT is also 1/64, as is TTHTHT. The issue is your brain only notices 2/64 strings containing only heads and tails, and doesn’t notice the 62/64 strings containing a mixture of heads and tails, even though the two are equally likely.
The confusion is that, with 1 flip, the odds of getting all heads or all tails are 2 strings out of 2 possibilities, or 100%. With 6 flips, the odds of getting all heads or all tails are 2 strings out of 64, or 1/32. So even though the odds of getting any given string is 1/64, the odds of getting a string that you *don’t notice* are over 95%. And that’s what causes the gambler’s fallacy: the disconnect between the actual probability, and the patterns that your brain notices.
Probability works by multiplying independent events together (so in your example, a coin coming up tails 6 times is 1/2* 1/2…).
However, once you flip the coin, the probability that it was tails is either 1 (if it was tails) or 0 (if it was not). So we can rewrite the likelihood that we get tails 5 times given we’ve gotten tails 5 times as 1*1*1*1*1, right? It’s happened, so there’s a 100% chance it’s happened. So the probability of 6 tails given 5 tails is 1 *1/2, or 1/2.
The fallacy is thinking that the past probability still has an effect on the future outcomes; so thinking that the probability of not getting heads in 6 coin tosses is (1- probability of all tails; 1-1/64, 63/64). However, given the 5 coin tosses already happened, it should be (1-probability of tails; 1-1/2, 1/2).
In addition: you are thinking “the odds of not getting heads”. However, what you’re really determining is “odds I get 5 tails and then 1 heads” vs. “odds I get 6 tails”. Odds you get 6 tails is 1/64, odds you get 5 tails and then heads is 1/64, the odds are ultimately 1:1.
You know it. The events are independent and it only feels dependent because of gambler’s fallacy.
Nothing about the coin flip will change the outcome of the other coin flips. There’s no magical force that will try to have a coin keep a running total of 50/50. That’s the gambler’s fallacy. You don’t flip two coins and have to have 1 head and 1 tail.
Russian roulette is a dependent event. You will die if you play it solo. If you flip coins, you can get heads every time. It’ll be statistically unlikely but possible.
Maybe ask why do you think the coin flips are dependent on each other because there is no reason besides you feeling it does and to quote a popular right wing phrase, “facts don’t care about your feelings.”
People are just bad at numbers and stats.
HHHHHH has the same chance as HHHHHT. These are the only two possibilities left since you’re already at HHHHH
The probability of flipping six tails in a row is completely different than the probability of the next flip being tails. You can’t mix up the two calculations. If you’ve already flipped five in a row, those flips are in the past and do not influence your next flip’s probability, which will always be 50/50. But if you’re about to make 6 flips, the chances of all 6 being tails )or heads) are indeed 1/64. You have to separate past from future. Same with the lottery. Playing yesterday’s winning number in tomorrow’s lottery gives you the same odds (low) as playing any other number, but the odds of the same number winning twice in a row in two future drawings are low*low. In one case you’re calculating the odds of a single drawing, and in another case you’re calculating the odds of a sequence of drawings. Not the same.
I’ts a 50-50 chance, every time. Flip the coin 1000 times and the results will come very close to the statistical norm.
The fallacy is because you’re confusing the odds of a single event with the odds of a series of events. You’re right that the odds of HHHHHH (six heads in a row) is pretty low. But they’re actually exactly the same as any other specific sequence of heads and tails. HHHHHH is exactly as likely as HTHTHT, or TTTHHH, or HHHHHT. Each is just one possible sequence of six coin flips. You’re kind of “bound to” get tails eventually, because as the number of coin flips increases, the number of unique combinations grows, but there’s still just one combo that’s all heads. So as you keep flipping, the odds *of your overall streak* go down. But the odds of the next coin flip, as a unique and independent event, don’t change. How could they?
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