Show that if a game is fair then the payoff odds must exactly match the odds against winning.
The result is a consequence of the bias rule, but
here is a direct argument using just the basic definitions.
| Let | y | = | payoff odds | |
| p | = | probability of winning | ||
| B | = | amount bet > 0 | ||
| g | = | odds against winning ( remember that 1/p = 1 + g ) | ||
| Then | ||||
| A win pays y * B ( by definition of y ) | ||||
| Let E = expectation = ( y * B ) * p + ( -B ) * ( 1 - p ) | ||||
| fair game | → | E | = | 0 | |
| → | yBp | = | B * ( 1 - p ) | ||
| → | yp | = | 1 - p | ||
| → | y | = | ( 1 / p ) - 1 | ||
| = | ( 1 + g ) - 1 = g | ||||
| i.e. fair game | → | payoff odds = odds against winning | |||