Set the odds to bake in a target margin

Question

You run a two-outcome book and believe the fair probabilities are $60\%$ for Team X and $40\%$ for Team Y. You want to post prices that carry a $5\%$ overround (your built-in margin).

What decimal odds should you offer on each side? What do you expect to hold on a balanced book?

Accepted Answer

De-vigging converts posted odds back to fair probabilities by dividing by the sum S. To set odds, run it forward: choose the target sum S = 1.05 and scale the fair probabilities up to it. Priced (implied) probabilities: pX = 0.60 	imes 1.05 = 0.63, \qquad pY = 0.40 	imes 1.05 = 0.42, which sum to 1.05, giving the wanted 5\% overround. Posted odds are the reciprocals: dX = \frac{1}{0.63} \approx 1.587, \qquad dY = \frac{1}{0.42} \approx 2.381. Expected hold on a balanced book: 1 - \frac{1}{S} = 1 - \frac{1}{1.05} = 4.76\% = \frac{	ext{overround}}{1+	ext{overround}} = \frac{0.05}{1.05}. A q…

Set the odds to bake in a target margin

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