Reading American, Decimal and Fractional Odds
Three notations describe the same thing, and each was shaped by the place that used it. American odds are written against a hundred: minus one fifty means a hundred and fifty risked to win a hundred, plus one fifty means a hundred risked to win a hundred and fifty. The hundred is a unit of account, not an instruction about stake size.
Decimal odds, the continental form, quote the total return per unit staked, so a price of two and a half returns two and a half units in all, of which one and a half is profit. Fractional odds, the British racecourse form, quote profit against stake, so the same price is written three to two.
Converting between them is arithmetic, not judgement. A decimal price is one plus the fraction. An American minus price divided by itself plus a hundred gives the implied probability; an American plus price is a hundred divided by the price plus a hundred. A price of minus one fifty implies sixty per cent; a price of plus one fifty implies forty per cent.
Add the implied probabilities on both sides of a two-way market and the total exceeds one hundred per cent. The excess is the margin described in the entry on the vigorish. It is the reason a set of prices is never a set of forecasts.