Something like a correlation for binary events, the odds ratio establishes the strength of association between two events.
The odds ratio for events A and B is defined as the ratio of {the odds of A in the presence of B} and {the odds of A in the absence of B}.
Two events are correlated when the odds ratio is positive and negatively correlated when the odds ratio is negative.
In terms of conditional probabilities, then:
$OR = (P(A|B) / P(¬A|B)) / (P(A|¬B) / P(¬A|¬B))$
This multiplies out in terms of joint probabilities to:
$OR = P(A,B) * P(¬A,¬B) / ( P(¬A, B) * P(A, ¬B) )$
Two events are {independent} if and only if their {odds ratio} equals {1}.
Q. Define the odds ratio for events A and B in terms of conditional probabilities.
A. $OR = \frac{\frac{P(A|B)}{P(¬A|B)}}{\frac{P(A|¬B)}{P(¬A|¬B)}}$
Q. Define the odds ratio for events A and B in terms of joint probabilities.
A. $OR = \frac{P(A,B)P(¬A,¬B)}{P(¬A, B)P(A, ¬B)}$
Q. What’s the mnemonic for remembering the equation for the odds ratio in terms of joint probabilities, $OR = \frac{P(A,B)P(¬A,¬B)}{P(¬A, B)P(A, ¬B)}$?
A. Concordant events divided by discordant events.