Probability and Odds

Risk, probability and odds can be confusing.

Risk has a broader, less specific and often refers to an expected loss. Risk can be absolute (no context) or relative (in context, for example sex). Risks don’t necessarily add up to one.

Probability is more specific and always has a value between 0 and 1. The probability of an event is:

\(Probability = \frac{EventHappening}{EventHappening + EventNotHappening} \)

Probabilities for and against add up to one; the probabilities for and against are each other’s complement.

The odds are for over against:

\(Odds = \frac{EventHappening}{EventNotHappening} \)

The odds ratio is the ratio of the odds for and against. The odds ratio for and against are each other reciprocal.

Of course, it is possible to ‘convert’ odds to probability.

To convert odds to probability:

\(Odds = \frac{a}{b}: Probability = \frac{a}{a+b} \)

Similarly for probability:

\(Probability = p: Odds = \frac{p}{1-p} \)

To illustrate on an example from the Titanic:

	Alive	Dead	Total
Female	308	154	462
Male	142	709	851
Total	450	863	1313

Risk

	Numerator	Denominator	Value	Comment
Overall Risk Death	863	1313	0.657
Risk Female Death	154	462	0.333	Not Complement of below
Risk Male Death	709	851	0.833	Not Complement of above
Relative Risk Male Death	0.833	0.333	2.5

Probability Female Death	154	863	0.178	Complement of below
Probability Male Death	709	863	0.822	Complement of above

And for the odds:

	Numerator	Denominator	Value	Comment
Female Odds	154	308	1/2
Male Odds	709	142	5/1
Odds ratio Males	5	0.5	10	Reciprocal of below
Odds ratio Females	0.5	5	0.1	Reciprocal of above