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.

Log Odds

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.

Table of Log Odds, Odds & Probabilities

 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.

Table of Gambling Odds, Odds, Prob & Log Odds

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

Categorical Data Analysis Course