Fractional odds of 3-to-2, what probability?

Question

A bookmaker quotes fractional odds of 3/2 ("three-to-two") on an outcome: stake \2 to win \3 profit, getting \5 back in total on a win. Solve in your head. What win probability makes this bet break even? Aim: under 5 seconds.

Accepted Answer

Fractional odds a/b mean you risk b to gain a, so a win returns a + b for every b staked. Break-even happens when the stake equals the expected return: p = \frac{b}{a + b} = \frac{	ext{stake}}{	ext{total return}}. For 3/2: p = \dfrac{2}{3 + 2} = \dfrac{2}{5} = 0.40, i.e. 40%. Equivalently, convert to decimal odds first: d = 1 + \frac{a}{b} = 1 + 1.5 = 2.5, and p = 1/d = 1/2.5 = 0.40. Same answer, either route. Takeaway. Fractional odds a/b imply probability \frac{b}{a+b}: stake over total return. Odds-against (a b) mean an underdog (probability below 50%); here 3/2 prices a 40% chance.

Fractional odds of 3-to-2, what probability?

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