You assess a binary event at 36%. What decimal odds are fair, what is the fractional equivalent, and what decimal price gives a backer a 5% edge per unit staked?

You assess a binary event at 36%. What decimal odds are fair, what is the fractional equivalent, and what decimal price gives a backer a 5% edge per unit staked?

Approach: Take the reciprocal of the probability for the decimal price, express the profit against the stake for the fractional form, then solve the expected profit condition for the price.

25/9. Fair decimal odds are the reciprocal of the probability, so 1/0.36 = 100/36 = 25/9, which is 2.7778. Fractional odds quote profit against stake, so they are (1 - p)/p = 0.64/0.36 = 16/9, written 16 to 9. A backer's expected profit per unit staked at decimal price d is p*d - 1, so a 5% edge needs d = 1.05/0.36, which is 2.9167 or 35/12. The check to run in the head is that the decimal and fractional forms always differ by exactly 1, since 16/9 + 1 = 25/9, so converting either way is one addition once the fraction is in lowest terms.

Follow-up: At a decimal price of 2.60 and a true probability of 36%, what is the expected loss per unit staked and how many bets before the loss exceeds one stake?

Key concepts: decimal odds, fractional odds, reciprocal, edge.