De-vig an American moneyline

Question

A US sportsbook posts a two-way moneyline: the favorite at -150 and the underdog at +130. What is the bookmaker's margin, and what are the vig-free probabilities?

Accepted Answer

Convert American odds to implied probabilities. For a negative line -O, the implied probability is 	frac{O}{O + 100}; for a positive line +O, it is 	frac{100}{O + 100}. p{fav} = \frac{150}{150 + 100} = 0.600, \qquad p{dog} = \frac{100}{130 + 100} = \frac{100}{230} \approx 0.4348. They sum to 1.0348, an overround of about 3.48%. That is the book's margin; betting both sides in proportion loses about 	frac{0.0348}{1.0348} \approx 3.4\%. De-vig (normalize to sum 1): \hat p{fav} = \frac{0.600}{1.0348} \approx 57.98\%, \qquad \hat p{dog} = \frac{0.4348}{1.0348} \approx 42.02\%. So the market's …

De-vig an American moneyline

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