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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

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>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?

If you flip a coin that you know is fair, then you would correctly expect even odds for heads and tails. You’d expect so even if you use a coin that has been flipped hundreds of times before. For a fair coin, the relation between past and future flips is entirely in your mind.

The probability of flipping 25 heads in a row is very low, but it’s only half as much as flipping 24 heads in a row, so the odds of flipping the 25th heads after already getting 24 is the same as getting heads once.

If you flip a coin three times, the chance of getting three heads in a row is 1/8, which makes sense because there are eight possible outcomes and they’re all equally likely.

The eight possible outcomes are

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

If after the second flip you’ve gotten two heads, then you eliminate all but the outcomes that had a tails in one of the first two spots.

HHH

HHT

~~HTH

HTT

THH

THT

TTH

TTT~~

There are only two possible outcomes remaining and they’re still equally probable, so there’s a 50% chance of getting HHH, and 50% chance of getting HHT.

The same logic works for 25 coin flips. All of the millions of possible outcomes with a tails in one of the first 24 flips have already been eliminated, leaving only two possible outcomes left.

Considering both gambler’s fallacy and logical paradox are outside the understanding of most 5 year-olds, I’ll use similar level terms to what were in the question. The gambler’s fallacy (and indeed most fallacies) exist because they seem like they should be sound based off of what society assumes. Or, based off of what society assumes to be true. The fact that you think it is a logical paradox shows that you are susceptible to this fallacy yourself and should probably avoid gambling. Not meant as an insult, everyone has some logical fallacy they believe in, it’s human nature.

The coinflip one you brought up only makes sense if the outcome of previous coin flips somehow change its balance. But in this case, you would have to know what it landed on every time it was dropped, flipped, or jangled in its life. This is a false scenario though so doesn’t matter, other than to show the absurdity of past flips altering the probability of the next one.

If, before a single flip, you were to bet on 25 flips in a row being heads, yes, those are some long odds. If, after 24 heads you stop and ask what the odds are the next one is heads, its still 50/50 because, well, past flips physically don’t change the coin so individual flips are still 50/50. It would be the same as saying “I’m going to flip this coin 25 times. What are the odds the last one is heads?” It’s 50% chance, from the first flip, that the coin will land on heads at number 25.

TL;DR Individual flips are always 50/50 and past events and perceived future events don’t change that. The fact that this seems illogical to some people is exactly why the fallacy exists in the first place.

Past results are not an indicator of future performance. The coin doesn’t know what happened.

Any specific combination of 25 flips would have the same odds. Singling out them all being the same only has psychological significance, not statistical significance.

Simply because you can just ignore the previous outcome. As a coin flip is completely random it will not care about what happened to it before, it will still have the 50% chance of being a head or a tail.

It’s because each flip of the coin isn’t affected by previous flips it’s still a 50/50 chance. The odds of getting heads after 24 tails is the same as getting tails after 24 tails 0.5^25

My brain won’t fully accept it either (if you really want your brain to hurt, look up the Monty Hall problem) but here’s the way I see it. There are 2 separate math problems here. The first one is “what are the odds of getting heads 25 times in a row” vs “what are the odds of getting heads just this one time”. You could then add 1 more flip and calculate the odds of getting HH. Then add 1 more flip, calculate the odds for HHH, and so forth. You’ll see the odds go way down pretty soon.

I don’t know whether this hurts or helps you, but imagine that you want the results of 25 coin flips to be HHHTTHTHTTHHHTTTHHTHTTHHT. You know the outcome can’t be a 50/50 chance. That’s a very specific pattern. And in fact has the exact same chance of happening as HHHHHHHHHHHHHHHHHHHHHHHHH.

Please let me know if this doesn’t help, so I never suggest it again, lol.

25 flips being heads is only unlikely because any of those 25 flips have a chance of being tails. After you’ve already flipped 24 heads, it’s no longer true that any of the 25 flips could be tails. Now there’s only one flip left that could be tails. The previous 24 flips are unable to be tails at this point. Your 25th flip can’t cause the already-set-in-stone 1st through 24th flips to change their results.

This assumes a fair coin, of course. After too many flips I would start to question the probability of living in a universe where this exact pattern happened versus living in a universe where the coin is biased and someone is cheating.

We’re the ones calling “25 heads in a row” something special. If your favorite pattern of choice was “3 heads, 2 tails, 5 heads, 1 tail, 1 heads, 7 tails, 1 head, 2 tails, 3 heads” then THAT would be the one seeming super unlikely.

The coin has no clue about the previous flips. It doesn’t say “oh all right I’ve been heads 24 times now, I’m tired, let’s do a tails for once”. It’s just dumb physics and every flip has the same rules. We’re the ones looking for patterns and being surprised if a pattern emerges. Every particular pattern is just as unlikely as any other particular pattern.

But one specific pattern of choice is of course much less likely than “any random old pattern”, because there are countless such random old patterns and only one “favorite” pattern. We lump the random patterns into one category and we put the special pattern alone in its own category. Then of course, “25 heads” is super unlikely compared to “anything else” because there is so much more of “anything else”.

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.