Overround from American moneyline odds

Question

A sportsbook quotes a two-outcome game in American (moneyline) odds: $-140$ on the favorite and $+120$ on the underdog. A negative number is how much you must stake to win $\$ 100 $; a positive number is how much you win on a$ $100$ stake.

Convert both to implied probabilities, then find the overround and the balanced-book hold.

Accepted Answer

Convert each price to an implied probability. Favorite at -140: p{	ext{fav}} = \frac{140}{140 + 100} = \frac{140}{240} = 0.5833. Underdog at +120: p{	ext{dog}} = \frac{100}{120 + 100} = \frac{100}{220} = 0.4545. Overround: S = 0.5833 + 0.4545 = 1.0379, \qquad 	ext{overround} = S - 1 = 3.79\%. Hold on a balanced book: 1 - \frac{1}{S} = 1 - \frac{1}{1.0379} = 3.65\% = \frac{0.0379}{1.0379}. De-vig for the fair split: dividing by S gives 0.5833/1.0379 = 0.562 and 0.4545/1.0379 = 0.438, so the book implies roughly a 56.2\% / 43.8\% true contest. For reference, -140 equals decimal odds 1 + 100/…

Overround from American moneyline odds

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