Suppose that the vig is not charged for a BUY bet unless you win. What would be the expectation and the house edge ?
| B | = | amount bet ( not counting the vig ) |
| V | = | the vig |
| p | = | P( win ) |
| g | = | odds against winning |
| E | = | expectation |
| H | = | house edge |
| E | = | ( gB - V ) p + ( - B ) ( 1 - p ) = gBp - Vp - B + Bp |
Now use 1 / p = 1 + g → 1 = p + gp → B = Bp + gBp
| E | = | - Vp | |
| H | = | | E | / bet = Vp / B | |
| V | = | B / 20 → E = ( - Bp / 20 ) and H = p / 20 |
| case 1 | Buying 4 or 10 | |
| p = 3 / ( 3 + 6 ) = 1 / 3 → H = 1 / 60 = 1.6666....% | ||
| case 2 | Buying 5 or 9 | |
| p = 4 / ( 4 + 6 ) = 2 / 5 → H = 2 / 100 = 2.00% | ||
| case 3 | Buying 6 or 8 | |
| p = 5 / ( 5 + 6 ) = 5 / 11 → H = 1 / 44 = 2.2727....% | ||
If a casino charges a vig only when you win, then
you might want to add the BUY bet to your list of "good" bets.