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Removing the margin

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A bookmaker quotes decimal odds of $1.8$ on team A and $2.1$ on team B in a match with no other outcomes. The raw implied probabilities do not sum to $1$ . After scaling both proportionally so they sum to $1$ , what is the probability of team A? Give an exact fraction.

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Hint

Take reciprocals of the odds, then divide each by their sum.

Solution

The raw implied probabilities are $\frac{5}{9}$ for A and $\frac{10}{21}$ for B.
Their sum is $\frac{35}{63} + \frac{30}{63} = \frac{65}{63}$ .
The scaled probability of A is $(\frac{35}{63}) \div (\frac{65}{63}) = \frac{7}{13}$ .

Answer: $\frac{7}{13}$

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