Overround, how much does the bookmaker keep?

Question

A bookmaker offers a two-outcome match (no draw) at decimal odds of $1.90$ on each side. Decimal odds of $1.90$ mean a $\$ 1 $stake returns$ $1.90$ total if it wins.

What is the overround, and what fraction of total stakes does the bookmaker expect to keep if the money is split so the book is balanced?

Accepted Answer

Implied probabilities. Decimal odds d correspond to implied probability 1/d: \frac{1}{1.90} = 0.5263 	ext{ each side.} Overround is the amount the implied probabilities exceed 1: S = 0.5263 + 0.5263 = 1.0526, \qquad 	ext{overround} = S - 1 = 5.26\%. A fair, vig-free market would sum to 1.00; the 5.26\% excess is the bookmaker's cushion. What they keep on a balanced book. Suppose bettors stake so that the payout is the same whichever side wins (the book is "balanced"). If total stakes are \1, the winning side is paid \1/S in total, so the bookmaker keeps 1 - \frac{1}{S} = 1 - \frac{1}{1.0526…

Overround, how much does the bookmaker keep?

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