# Do past probability results of a fair coin affect the present probability?

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!

• en.m.wikipedia.org/wiki/Gambler%27s_fallacy Sep 4 '20 at 19:09
• 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. Sep 4 '20 at 19:10
• 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 (${50\%}$) Sep 4 '20 at 19:19
• 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.
– lulu
Sep 4 '20 at 19:19
• 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. Sep 4 '20 at 19:29

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.