Nice avatar!
Quote:
Using the same logic, a "streak" of 1 head (WLOG) would be (1/2)*1,000,000 = 500k to 1 on. That seems way too short.
That wouldn't be 500k to 1, it would be 500k as in (50 million)% lol. The reason that happens is you're overcounting the ways to get more than one success. The actual chance would be 1-.5^1000000 as in pretty much 100%.
When it comes to a streak of 100, it's tricky because the series of 100 aren't independent. If you get a tails on the 50th flip, that doesn't only kill one streak, it prevents any streak from starting earlier than flip #51.
If we were talking about <200 flips then there would be an amazingly simple inclusion-exclusion formula, but with 1M flips the possibility of separate successful streaks thwarts it.
Instead, you can use one of the formulas provided in
this paper, or since you're talking about p=.5 flips, there is also the formula involving
higher-order Fibonacci numbers. Yet another solution is a matrix method (markov chains). You've got options!
Anyway, if it has to be Heads, the answer is 1 in 2.53555 * 10^24
If it can be H or T, then since P(H) is so low, P(H or T) is basically 2x that, so 1 in 1.268 * 10^24
Last edited by heehaww; 05-30-2020 at 06:23 PM.
Reason: Mistype: 2nd decimal place was wrong