I know it’s true, it’s just something that seems hard to wrap my head around. How is this not a logical paradox?
In: Mathematics
By the time you’ve hit 24 heads in a row you’re already in the universe where the 1 in 16777216 chance happened, and in fact there is nothing special about it. The odds of *any* combination of heads and tails also has a 1 in 16777216 chance of happening in that exact order. You’ve thrown 24 heads in a row? Well, now there’s a 100% chance you have 24 heads in a row, you already determined that; but there sequence HTHHTHTTTTHTTTTHHHHHTHH is just as likely even if it doesn’t seem as special as 24 heads.
The 25th coin flip is simply determining if you’re entering the universe where the sequence you already have produced ends in a heads or tails. Two options: HHHHHHHHHHHHHHHHHHHHHHHH or HHHHHHHHHHHHHHHHHHHHHHHT. Those are your only two options at this point: and they are equally likely. 50:50
Edit: With the assumption the coin is fair to begin with. In the real world if someone *actually* manages to flip 24 heads in a row I’d stop thinking about the gambler’s fallacy and shift towards thinking about potential fraud.
>but **if you flip heads 24 times in a row**, the 25th flip still has odds of exactly 0.5 heads. Isn’t there something logically weird about that?
“**If**” you do that it has already happend. The chance of *having flipped* heads 24 times in a row is 100% if it is your base assumption. That is vastly different of the chance of flipping heads 24 times in a row in your future 24 flips.
When you’re looking at probabilities, you’re looking at the odds of an event happening from some initial condition. The odds of getting 25 heads in a row starting from 0 is very low. However, the odds of getting the 25th head after already getting 24 in a row is 50/50.
To put it another way, the probability of anything that actually happened is always 100%. When you’re going for the 25th flip, you always have 24 flips done, so the odds of the first 24 flips in that situation are 100%, and only the 25th flip matters.
To the coin, each flip is an independent with no memory of the previous events even though to us, it may seem like there’s a pattern.
i work in a casino, and we have an obligation to disprove gambler’s fallacy. we have a massive amount of resources *and* a GameSense Advisor on site every day to do this, and regular staff are trained in (at the very least) the basics on how to dispel myths.
there are two types of gamblers. those who *understand* the odds, and those who don’t, and those who don’t, *really cannot*.
it doesn’t matter how hard you try to prove it. it doesn’t matter if you have statistical models and factual proof of payouts being *entirely random*, they will still believe they’re ‘due a win’ or that ‘luck is on my side, you watch’.
for slots, the most common misconception is the ‘jelly beans in a jar’ belief; that there’s 9999 black jellybeans and 1 red one and that every time they pull the handle, the number of black jellybeans goes from 9999 to 9998, and so if they spin long enough, they’ll go from 9999:1 to 1:1, and they’ll win. when in *fact*, it’s *always* 9999:1 and the only thing spinning the reels does is shake that jar full of 10000 jellybeans up.
the belief is pervasive. i’d like to say that all it takes is a few really bad losses for someone to figure it out, but that is almost never the case. the worse their losses, the more convinced they become that ‘the next time’ or ‘this machine’ or ‘this table’ will be the one to help them recoup their losses.
gamblers also never talk about *how much money* they spent to get that big win. *sure,* you might’ve just pulled down a 10k jackpot, but i watched you sit at that dollar spot hitting max bet for *five hours*. and at about 100$ per spin, and at *about* 20 spins per *minute* … sure you’ll have big and small wins leading up to that 10k, but in nearly every case, that ‘big’ jackpot brings a player close to breaking even, and *rarely in the win column.*
The odds of the 25th coin flip are just 0.5 if u don’t look at the 24 flips before that and if u dont care about the 24 before, there is no difference between throwing one and the 25th.
Flipping 25 heads in a row (24 heads in a row followed by 1 more heads) has *exactly* the same odds as flipping 24 heads in a row followed by 1 tails.
So after you’ve flipped 24 heads in a row, it should make sense that the next flip has equal odds of being heads or tails.
Flipping a coin doesn’t change the coin in any way.
Assume a fair coin, which isn’t two headed and doesn’t favor one side. When you flip it there’s a 50% chance you get heads. No matter what you get, the next time you flip it it’s still a 50% chance because it’s the same coin. Nothing about it has changed.
Looking at it from a math perspective, the chance to get 25 heads in a row is 1 : 0.5^25 . But the chance to have gotten 24 heads in a row, if you already got 24 heads in a row, is 1^24, or just 1. So the chance to get that 25th heads is 1X0.5^1, or 0.5.
Probability doesn’t work backwards through time. The probability of anything that happened, having happened, is always 100% after the fact.
The chance that the 25th flip with also be heads is dependent on another factor though: am I betting money on this flip? If ‘yes’ it will be tails, if ‘no’, it will be heads.
You’re talking about the odds of a set of things happening vs a single instance.
An example
the chances of everything that led to the evolution of man vs the chances I’m going to go get a coffee.
Damn. All the things that lead to my creation and now I’m gonna go get a coffee. Can’t wrap your head around that?
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