De-vig a two-way betting market

Question

A bookmaker offers decimal odds of 1.25 on the favorite and 4.00 on the underdog (two-outcome event, no draw). What is the bookmaker's margin, and what are the vig-free probabilities of each outcome?

Accepted Answer

Implied probabilities are reciprocals of decimal odds: p{fav} = \frac{1}{1.25} = 0.80, \qquad p{dog} = \frac{1}{4.00} = 0.25. They sum to 1.05, an overround of 5%. That excess is the vig: the bookmaker has priced both outcomes slightly "too likely," so betting either side at these prices costs you about the margin in EV. (Equivalently: bet both sides in proportion and you lock in a loss of \approx 	frac{0.05}{1.05} \approx 4.8\%.) De-vigging (multiplicative method): rescale so probabilities sum to 1: \hat{p}{fav} = \frac{0.80}{1.05} = \frac{16}{21} \approx 76.2\%, \qquad \hat{p}{dog} = \frac…

De-vig a two-way betting market

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