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Bookmaker locked profit

Market makingMedium

On the $3$ outcomes A, B and C of an event, a bookmaker offers decimal odds of $1.6$ , $4$ and $6$ , and a winning bet returns stake times odds. Bettors place stakes of $1500$ on A, $600$ on B and $400$ on C, and exactly one outcome occurs. What profit does the bookmaker make, and does it depend on the outcome? Give the profit.

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Hint

Compute the payout under each outcome.

Solution

If A wins the payout is $1500 \times 1.6 = 2400$ , if B wins it is $600 \times 4 = 2400$ , and if C wins it is $400 \times 6 = 2400$ .
The total staked is $1500 + 600 + 400 = 2500$ .
The profit is the same in every case: $2500 - 2400 = 100$ .

Answer: $100$

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