From your probability to fair odds

Question

You estimate an event is 30\% likely. What are the fair decimal, American, and fractional odds, and what is the lowest decimal price you should accept to bet on it?

Accepted Answer

Odds are just a probability restated as a price, so invert the conversions. Decimal. Fair decimal odds return 1/p per unit staked: d = \frac{1}{0.30} = 3.33. At exactly 3.33 the bet is break-even (zero expected value); you only have positive EV at a price above 3.33. American. Since p < 0.5 this is an underdog, so plus odds +x with p = \frac{100}{x + 100}. Solving, x = \frac{100(1 - p)}{p} = \frac{100 	imes 0.70}{0.30} = 233, so +233. Fractional. Odds a/b satisfy p = \frac{b}{a + b}, so \frac{a}{b} = \frac{1 - p}{p} = \frac{0.70}{0.30} = \frac{7}{3}. Check: p = \frac{3}{7 + 3} = 0.30, and th…

From your probability to fair odds

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