Odds, in full
Strictly, odds compare favourable outcomes to unfavourable ones, and probability compares favourable outcomes to all outcomes. Rolling a 6 on a d6 gives one favourable face and five unfavourable, so the odds against are 5 to 1 while the probability is 1/6. The two convert cleanly. If the odds against are a to b then the probability is b/(a+b), and if the probability is p then the odds against are (1-p) to p.
Lotteries almost never use that strict form. They quote odds as 1 in N, which is really a probability written as a reciprocal, so 1 in 13,983,816 means a probability of 1/13,983,816. Written strictly, the odds against would be 13,983,815 to 1. At lottery scale the difference is invisible. At short odds it is not: a coin is 1 in 2 by the lottery convention and 1 to 1 by the strict one.
One further convention causes trouble in marketing. Overall odds means the chance of winning any prize at all rather than the top one, and the figure is dominated by the smallest tier, so it never describes a jackpot. It has its own entry here, because the two numbers get printed beside each other and read as though they measured the same event.
- Probability
- Odds against, Chance
A worked example
Jackpot odds of 1 in 13,983,816, converted: 1 divided by 13,983,816 is 0.0000000715, which is 0.0000072 percent once multiplied by 100. Written as strict odds against, the same chance is 13,983,815 to 1. At the other end of the scale, the odds against rolling a 6 on a d6 are 5 to 1, a probability of 0.1667, or 16.67 percent.
See also
Where it comes up on this site
Odds: frequently asked
No, though everyday usage blurs them. Probability sets favourable outcomes against the total. Odds set favourable against unfavourable. A probability of 1/6 is 5 to 1 odds against.
About 0.0000072 percent. Divide 1 by 13,983,816, then multiply by 100.
Because they count every tier, including the smallest. A figure like 1 in 9 and a jackpot figure in the millions describe two different events.