Interesting indeed. "On average" is understandable for the layman, but I find it vague. I gather it means exactly 50% odds (to get a sequence of 11 losses at 50% chance per loss). If you can say that way. I'd suppose graphically it comes from the bell-shaped distribution curve, exactly half of which is dissected at the 2046 plays.
I'm sorry, I was being lazy. I mean the expected number of rolls. ie. the sum of the product of the number of rolls and the probability of it taking that many rolls.
Also, I find it unbelievable that getting exactly 2 heads in a row for a fair coin flip takes on average 6 flips! "On average" again is supposed to mean 50% probability. Need to twist my head around that for a while.
How's this for weird? A plot of probability against number of rolls. There's a 0.25 probability that it will take 2 rolls to get 2 heads, etc:
The probability of it taking n rolls is:
fib(n-2) / 2^n
where fib() is the Fibonacci sequence!
fib(0) = fib(1) = 1
fib(2) = 2
fib(3) = 3
fib(4) = 5
fib(n) = fib(n-1) + fib(n-2)
And you thought Mr. Fibonacci was just for the wackos doing technical analysis of the Bitcoin price?
Edit: weird because of the step in the graph. The probabilities that it will take 3 or 4 tosses to get 2 heads in a row are the same:
2: 1/4
3: 1/8 (THH)
4: 2/16 (TTHH and HTHH)
5: 3/32 (HTTHH, HTTHH, TTTHH)
6: 5/64
7: 8/128
8: 13/256
