Suppose that the probability to win a certain game is
exactly 1 / 10, and the payoff odds are ten to one.
Is this a fair game ?
We will solve the problem two different ways.
| Let B | = | size of Bet | |
| W | = | amount you stand to win | |
| p | = | P( win ) = 1 / 10 | |
| g | = | odds against winning |
Method 1 -- See if the payoff odds match the odds against winning
1 / p = 1 + g → 10 = 1 + g → g = 9 = 9 / 1
So, a fair game would require paying off at 9 to 1.
Thus, the answer is No, the player has an advantage.
Method 2 -- Use the bias rule, where we compare W to gB
Since the payoff is 10 to 1, W = 10B; and,
since g = 9, W = 10B
> 9B = gB.
W > gB → game is biased in player's favor.
We've shown ( in two different ways ) that the game is unfair.
If we wanted a fair contest, what should the payoff odds really be ?
First, let's verify that the current game gives the player a positive expectation.
| E | = | Wp + ( - B ) ( 1 - p ) | |
| = | ( 10B ) ( 1 / 10 ) + ( - B ) ( 9 / 10 ) | ||
| = | B / 10 > 0 | ||
This is a better deal than you will find in a casino, where E is almost always negative.
| Let x | = | payoff odds for a fair game | |
| Then E | = | ( xB ) ( 1 / 10 ) + ( - B ) ( 9 / 10 ) |
fair game → E = 0 → xB / 10 = 9B / 10 → x = 9
To make the game fair, the payoff odds should only be 9 to 1.