There is a betting strategy for coinflips, called Martingale, in which the player should pick heads or tails and keep betting on it (doubling their bet on every loss) to eventually make a win when it appears. In theory this is watertight, but with one caveat: there can be very long strings of losses, and the player runs out of money, and looses it all.
Now, some people wait until they have seen a very long string of outcomes for one side, and start betting at the other side, with the idea that has a larger chance now. I know that mathematicly this is nonsense, both sides still have 50% chance no matter what happend before.
Although I can understand a dice/coin/ball has no notion of past/future, so will always remain 50% chance, I never quite understanded the following. You can proof there is only a very slight possibility of me ever witnessing 50 heads in a row during my lifetime. So after 49 spins, I will either observe something extraordinary (50 heads) or something less special (first tail), how can that not effect chance?
Maybe the answer is obvious, that for math/statistics it doesnt matter whether someone is observing the event or not, but please enlighten me.
I would say that if it's the case that you have already seen 49 heads in a row (history), you have already witnessed something extremely unlikely.
What you presently see on the $50$th flip, heads or tails, is a "toss-up" (excuse the pun!): the outcome of which is hardly extraordinary: tails or heads.
You're right, Joshua. The probability of getting $49$ heads followed by a head $\left(\dfrac 1{2}\right)^{50}$ is equal to that of getting $49$ heads, followed by a tails: $\left(\dfrac 1{2}\right)^{49} \cdot \dfrac 12.\;$ Both extremely unlikely events, taken as a whole.