We have seen how to interpret the betting odds or moneyline. Take, for example, the NBA odds for one of tonight’s matches.
| Team | Moneyline |
| Washington Wizards | +340 |
| Boston Celtics | -450 |
If you place $100 on the Wizards, and they win, you get $440, suggesting a profit of $340 upon winning (and you lose $100 if they lose).
On the other hand, if you bet $450 on the Celtics, and they win, you get $550 (and a loss of $450 if they lose). Your profit for winning is $100.
Implied probability
Let’s apply the expected value concept to this problem. Let p be the probability of winning the prize upon betting on the Wizards. For this to be a fair gamble,
the expected value = p x 340 – (1-p) x 100 = 0
p = 100/440 = 0.227 or 22.7%; this is an implied probability of the bet.
Let q be the probability for the Celtics.
the expected value = q x 100 – (1-q) x 450 = 0
q = 450/550 = 0.818 or 81.8%
Suppose you add the two probabilities, p + q = 104.5%. It is more than 100%, suggesting they are not the actual probabilities (fair odds) of winning the (NBA) game. Since the actual win probabilities of teams must add up to 100%, the sum p + q must be lower than that obtained in the expected value calculations. Therefore, at least one of the expected values must be < 0.
104.5% may be understood as putting $104.5 the money at risk to get $100 back. The difference (104.5 – 100)/104.5 is the bookie’s edge built into the bet.
