Actually the title of this unit is Odds & Probabilities not Odds & Ends. However, Odds & Ends is catchier.

Odds are one of the ways that people use to express the likelihood of events. When you hear someone say that the odds are 3-to-1, they are saying that the probability of an event occurring is three times greater than the probability of the event not occurring. An even shorter way of expressing this is just to say that the odds equal 3, which implies 3-to-1. When the odds are 3 that means that the probability of the event is .75 and the probability of event not occurring is .25, i.e., 3-to-1.

If the odds are 1-to-3, we could also say that the odds are .3333. The probability of the event is .25 and the probability of ~event is .75. The pair of odds 3 and .3333 are reciprocals of each other, thus 1/3 ≅ .3333 and 1/.3333 ≅ 3. We could call them a reciprocal pair.

Sometimes you will hear someone say that the odds are 3-to-2. This is equivalent to saying that the odds are 1.5-to-1 or just 1.5, for short. The probability of the event is .6. If the odds were 2-to-3, we could say the odds were .6667 and that the probability was .4. The odds 1.5 and .6667 form a reciprocal pair (1/1.5 ≅ .6667 and 1/.6667 ≅ 1.5).

When the odds are 1-to-1, or just 1 for short, the probability of the event is .5.

You will sometimes hear these kinds of probabilities in terms of chance, that is, 1 chance in 4, or just 1-in-4. It is easy to convert from odds to chance. If the odds are 1-to-3 then the chance is 1-in-4, this because the total for 1+3 from the 1-to-3 is 4 which becomes the last part of the chance statement.

The log odds of 1 are 0, i.e., ln(1-to-1) = ln(1) = 0.

Please note the natural log of zero is undefined.

log odds prob odds 4.0 54.5982 0.9820 3.5 33.1155 0.9707 3.0 20.0855 0.9526 2.5 12.1825 0.9241 2.0 7.3891 0.8808 1.5 4.4817 0.8176 1.0 2.7183 0.7311 0.9 2.4596 0.7109 0.8 2.2255 0.6900 0.7 2.0138 0.6682 0.6 1.8221 0.6457 0.5 1.6487 0.6225 0.4 1.4918 0.5987 0.3 1.3499 0.5744 0.2 1.2214 0.5498 0.1 1.1052 0.5250 0.0 1.0000 0.5000 -0.1 0.9048 0.4750 -0.2 0.8187 0.4502 -0.3 0.7408 0.4256 -0.4 0.6703 0.4013 -0.5 0.6065 0.3775 -0.6 0.5488 0.3543 -0.7 0.4966 0.3318 -0.8 0.4493 0.3100 -0.9 0.4066 0.2891 -1.0 0.3679 0.2689 -1.5 0.2231 0.1824 -2.0 0.1353 0.1192 -2.5 0.0821 0.0759 -3.0 0.0498 0.0474 -3.5 0.0302 0.0293 -4.0 0.0183 0.0180

Just for reference we will display some of the more common gambling odds for comparison.

vegas chance odds prob log odds odds 10to1 10in11 10.0000 0.9091 2.3026 9to1 9in10 9.0000 0.9000 2.1972 8to1 8in9 8.0000 0.8889 2.0794 7to1 7in8 7.0000 0.8750 1.9459 6to1 6in7 6.0000 0.8571 1.7918 5to1 5in6 5.0000 0.8333 1.6094 4to1 4in5 4.0000 0.8000 1.3863 3to1 3in4 3.0000 0.7500 1.0986 2to1 2in3 2.0000 0.6667 0.6931 1to1 1in2 1.0000 0.5000 0.0000 1to2 1in3 0.5000 0.3333 -0.6931 1to3 1in4 0.3333 0.2500 -1.0987 1to4 1in5 0.2500 0.2000 -1.3863 1to5 1in6 0.2000 0.1667 -1.6094 1to6 1in7 0.1667 0.1429 -1.7916 1to7 1in8 0.1429 0.1250 -1.9456 1to8 1in9 0.1250 0.1111 -2.0794 1to9 1in10 0.1111 0.1000 -2.1973 1to10 1in11 0.1000 0.0909 -2.3026

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