Suppose you are playing a game where a fair coin is to be flipped 1...

Suppose you are playing a game where a fair coin is to be flipped 1,000 times and you are allowed to bet on heads or tails. The coin lands on heads for the first flip, then astonishingly lands on tails for the next 500 flips. Since the coin theoretically will tend to a 50/50 distribution, should you start betting heads? How is betting heads in this scenario different from a game with the same coin, same number of flips, and placing a bet on turn 2 that the next 501 flips will all be tails? You're making essentially the same bet with retroactive knowledge, but that knowledge doesn't change the probability that the coin will land on the same side 501 times in a row.

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en.wikipedia.org/wiki/Gambler's_fallacy#Why_the_probability_is_1.2F2_for_a_fair_coin
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You're not betting that it will land on the same side 501 in a row - that was a surprise to you. You're only betting that the next flip will be tails. It's 50/50 each time regardless of what happened before.

those are different sets entirely

P(A) isn't the same as P(A|B)

actually i just thought of a really good analogy if you're actually having trouble with this and not just trolling

consider that the odds of an American being struck by lightning are just short of one in a million. if someone offers you million to one odds that they can stand in the middle of a field during a heavy storm and not get struck by lightning, should you take the bet?

the situation here is similar; something has happened that makes a certain outcome much more likely than it was before. already having had 500 flips in a row land on tails makes it a lot easier to get to the 501st flip on the next try, just the same as standing in the middle of a field during a storm makes it a lot easier to be struck by lightning.

wut

Baka Ritz baka Ritz la la la

EXPLOSION

easy

betting heads is exactly the same as betting tails

therefore betting heads is no less wrong than betting tails

so go ahead and bet heads just in case the "fair coin" assertion turned out to be incorrect

en.wikipedia.org/wiki/Gambler's_fallacy#Why_the_probability_is_1.2F2_for_a_fair_coin

What if the game is that the coin will be flipped 500,000,000 times and 250,000,000 of the first 275,000,000 flips were heads. Here the law of large numbers applies, should you bet tails?

>then astonishingly lands on tails
stopped reading there

If the flips turn out as stated, then something fishy is going on.

"Fat Tony" (Nassim Taleb character) would advise making your excuses and leaving.

The next flip is always 1/2

As the page explains, while 250,000,000 flips being always heads is statistically absurd, it doesn't affect the next flip

However...

>In most illustrations of the gambler's fallacy and the reversed gambler's fallacy, the trial (e.g. flipping a coin) is assumed to be fair. In practice, this assumption may not hold.

>For example, if one flips a fair coin 21 times, then the probability of 21 heads is 1 in 2,097,152 (above). If the coin is fair, then the probability of the next flip being heads is 1/2. However, because the odds of flipping 21 heads in a row are so slim, it may well be that the coin is somehow biased towards landing on heads, or that it is being controlled by hidden magnets, or similar.[3] In this case, the smart bet is "heads" because Bayesian inference from the empirical evidence — 21 "heads" in a row — suggests that the coin is likely to be biased toward "heads", contradicting the general assumption that the coin is fair.

TL;DR Since it's highly unlikely, the coin is probably rigged. But if it isn't it's still 50/50

Man this 'guide' went from the ramblings of a /pol/-tier make supremacy autist to superstitious /x/ garbage very quickly

It is if A and B are independent. As they are in this case.

You bet heads because shit is rigged

LOLe....this is science board

My stats teacher covered this very question and the answer is bet tails... the coin is somehow biased.

Every coin flip is independent from the others. (i.e. a coin won't change based on the amount of heads/tails previously thrown.)
Odds are always 50/50.

>It's 50/50 each time regardless of what happened before.

yep.

if every coin toss is independant of the previous (which is true) then it does not matter if you bet on heads or tails.

assuming it is a fair coin toss (50/50) then you are basically wagering on the odds = (0.5)^n where n is the number of throws.

because the events are independant there is no advantage to wagering heads even if the tosses resulted in 1000000 heads

>a really good analogy
that's not a good analogy at all. there are priors at play (your position, the weather) that are not taken into account when formulating that 1 in a million statistics.

in other words
P(struck_by_lighting) = 1/1000000
P(struck_by_lighting | in_a_field, lightning_storm_going_on) >>> 1/1000000

there are no such extenuating priors in the case of the coin flip. each trial is completely independent.

>each trial is completely independent.
this is true.

these are not dependant events.

each coin flip is 100% independant of the last flip

Heres an example:

I flip a coin once. I get heads. Should I bet on heads again because that is literally the only result that has occurred?

No, it doesnt matter because the next flip entirely is independent of the first

??? there definitely are extenuating priors in the situation in the OP. it's 3*10^150 more likely to get a 501st heads result in a row immediately after getting 500 heads results than it is to just get 501 heads in a row.

>most professors will give a student an A in any semester for 5000 dollars

lol ya risking a 70k (at fucking minimum) job with possible tenure, probably never being employed in academia again, and being humiliated on local if not national news is totally worth 5k.

Suppose this scenario were to actually occur. The odds of a coin that yields truly independent outcomes every flip only flipping for one side so many times is already massively improbable. However, given that the coin us able of independent outcomes, the bet you make can disregard previous events, no matter how absurd. Outside of a simulation though, you bet with the side of the coin that has managed to be flipped for 500 times due to the likelihood that there is no independence and so other factor is in effect which is unknown.

Any outcome of sequence of flips is equally probable, just like the lotto's chance of coming up 1 2 3 4 5 6 7 has the same probability of spacing out the numbers.

there are no extenuating priors because that's not the way the universe works provided you are 100% guaranteed at the beginning of the game that you do indeed have a fair coin.

No coin flip can be truly random.
Anything mechanical is a chaotic system. Flipping the coin in the middle will result in in landing with whatever face up as when it was flipped without spinning. Anything else is a variation on the initial condition which theoretically can be calculated to determine how it will land. If a coin/ roulette wheel/ lotto ball machine/ etc. consistently lands a certain way, it is better to bet with the trend, since the actuator of motion has a tendency to set initial conditions that result in that outcome.

tl;dr most of these metaphors don't really apply since a "truly random" mechanical outcome generator is an idealized machine.

This is comedy gold.

no, dipshit, this has nothing to do with having a fair game. I don't know how you keep misreading this.

Literally all I'm saying is that you're more likely to get one head than you are to get 500.

wow how interesting no one here knew that