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I decide to play a game of heads or tails with an absolute "fair" coin $1,000,000$ times. Suppose (however unlikely) I get $999,999$ heads in a row!

If I get heads on the 1 millionth toss, I win a prize! Should the past coin tosses influence my present decision?

Logically, the last toss is still $50$% to get heads... but can I allow my mind to bias the probability and favor heads over tails?

Thanks for any insights!

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    $\begingroup$ en.m.wikipedia.org/wiki/Gambler%27s_fallacy $\endgroup$ Sep 4 '20 at 19:09
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    $\begingroup$ It’s tricky. Practically speaking the issue is that if you ever found yourself in this situation then it’s extremely likely that the coin isn’t actually fair. To do a complete analysis involves an analysis of why you think you know the coin is fair which is underspecified by the problem. $\endgroup$ Sep 4 '20 at 19:10
  • $\begingroup$ In addition to my answer - indeed as @QiaochuYuan has mentioned, if this happened in real life we would obviously start to question the fairness of the coin. But if we (theoretically) knew the coin was absolutely fair - as my answer states - the probability remains exactly the same on the last go (${50\%}$) $\endgroup$ Sep 4 '20 at 19:19
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    $\begingroup$ Say you believe that there is a one in a million chance that someone slipped you an unfair coin. Then that possibility will loom very large given the preposterously improbable event you observe. $\endgroup$
    – lulu
    Sep 4 '20 at 19:19
  • $\begingroup$ Call heads. If the coin is fair, it doesn't matter what you do, and if the coin is biassed, it's overwhelmingly likely that the next toss will be heads. $\endgroup$
    – saulspatz
    Sep 4 '20 at 19:29
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The answer since you stated the coin was known to be fair:

If you get a huge number of heads in a row - the chance you get another heads on the next toss is still ${50\%}$. The thing is - those ${999,999}$ heads in a row have already happened, and the events are independent. [Basic conditional probability]: for independent events ${A,B}$ we know that

$${P(B|A)=P(B)}$$

In this case - ${B}$ is "I get a heads next toss" and $A$ is "I just got ${999,999}$ heads in a row".

If you ask "what's the chance I get ${1,000,000}$ heads in a row?" without having done any tosses already, then the answer is a small number. Nothing has happened yet. It may seem odd since your intuition tells you "come on, the last one must be a tails!" but it's still the exact same probability, since those tosses have now already occurred.

For a real world coin - if you throw it a lot of times in a row - it technically I guess could become slightly "weighted" overtime and have a slight bias as a result - but in a theoretically perfect coin - the probability remains the same. Also if you did get that many heads in a row with a real world coin - we would obviously start to question whether that coin really was fair.

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