I was watching tossing a coin, and saw $100$ times in a row there was tails.
Now I need to make my bet – and people/science(?) say, that the probability I will have heads is no more than $50\%$.
But according to Bernoulli's formula, $101$ tails in a row is extremely rare, so based on this Bernoulli's formula I should make a conclusion that $101$-tails-in-a-row is very small possibility, so I shall surely stake on heads.
How Bernoulli's formula (probability to get $k$ tails on $n$ trials, $n=k$ in our case) correspond to this "$50/50$" chance when "predicting" results after $100$-tails-in-a-row event? I mean $C_n^k\cdot p^k\cdot (1-p)^{n-k}$, in our case $n=k$ (number of trials $n$ = number of "successes" $k$), $p$(heads)$=p$(tails)=$\frac{1}{2}$.
So probability of $101$-tails-in-a-row is $\frac{1}{2}$ to $101$th power which gives $3.944304526105059\cdot 10^{-31}$ !!!
Why saying "Whoa, chances for next heads is just $50\%$ – regular $50/50$" I shall ignore that chances of that happening are close to zero??? Because chances for next tails in the GIVEN situation are equal to chances of Bernoulli's $101$-in-a-row = $3.944304526105059\cdot 10^{-31}$ which means that chances of heads = $1-3.944304526105059\cdot 10^{-31 }= $ almost $100%$ ???
Yes, probability and trillions of trials may have in stock for us all kinds of hard-to-believe events, and many $101$-in-a-row or $1001$-in-a-row – infinitely many of them, but probability of all these events is extremely small – what "mysterious something" compensates for such small probability of such events and makes it $50/50$ for next coin toss?
Then again, if I witnessed $101$-in-a-row once in my life, it seems logical to think that because probability of $101$-in-a-row is small, then my chances to see it again in my lifetime is small – I will "surely" never see it again, so next time I see $100$-in-a-row, I can stake on heads and have probability more than $50\%$?
I know about Gambler's fallacy and stuff, but I cannot match it with Bernoulli's formula of probability and cannot understand it logically, reasonably and from practical point of view, not just cram/remember theory – I want to understand it deeper.
P.S. And what about this Bernoulli's probability versus "coin has no memory, so after $100$-in-a-row tails chances of tails again are $50\%$"?
Maybe we should differentiate different things:
- Probability of single event – tossing coin just once. And that's all, observations finished.
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Probability of heads/tails – as seen averaged over millions of trials – it is close to $50/50$ (such $100$-in-a-row already averaged out).
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Probability of concrete complex event – where we speak not about average probability of tails, but exactly about probability of getting tails after $100$-tails-in-a-row. Yes, over millions of trials probability is $50/50$, but exactly today, here, at this place and moment and in this event – it is not $50/50$, it is $1:10^{31}$.
Best Answer
The probablilty of "101 tails in a row" is small.
The probability of "100 tails in a row and then heads" is also small. In fact it is exactly the same.
Once you've seen 100 tails, you know that the next throw you make is going to end you in one of those two cases, so you cannot use the smallness of each of those probabilities to conclude that one is more likely to happen than the other.
Basically you're conflating the absoulte probability of a long streak of tails with the conditional probability of another tails, given that you already know that most of the extremely unprobable event has happened already.
(As noted in comments, if you see 100 tails in a row, you should probably very carefully consider if you still believe your implicit assumptions that the coin is fair and the throws are independent -- and if you don't believe that, the simple probabilistic analysis is not valid anyway).