Bet You a Million
19 Tails before the first Head means a prize of
£1,048,576
And this happens "on average" once every 1048576 games (2^20)
The answer is that any amount of stake value will still produce a
profit in the long run, provided Gambler's Ruin is avoided.
P(£2) = 1/2
P(£4) = (1/2)^2
P(£8) = (1/2)^3 etc.
The average winnings per game in the long run is :
2 x (1/2) + 4 x (1/2)^2 + 8 x (1/2)^3 + . . . . .
or 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + . . . . .