Conditional Probability is Not Commutative: Formulas and Examples

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Confusing the “given” event (the event that you assume to have occurred) with the combined event (for which you are calculating probability) is a common pitfall with conditional probability.

Probability of the Intersection Sum of Two Dice: Image by Mike DeHaan

Probability of the Intersection Sum of Two Dice: Image by Mike DeHaan

A Recap of Conditional Probability

Recall that “the conditional probability of event ‘A’, given that event ‘B’ has occurred, is calculated as the probability that both ‘A’ and ‘B’ occur, divided by the probability of ‘B’ alone”. The equation is “P(A|B) = P(A∩B) / P(B)”.

Just to be clear: “P(A|B)” is “the conditional probability of ‘A’, given ‘B’ occurred”; while “P(A∩B)” is “the probability of the intersection of outcomes for both ‘A’ and ‘B’, ie that both ‘A’ and ‘B’ happened”.

Combined and Independent Events

Imagine tossing a “fair” (unweighted, unshaved) coin twice. Each toss is independent; either could result in heads or tails, and the first toss does not influence the second.

However, even independent events can have a combined outcome. If there were a wager that both independent coin tosses would land heads, then the probability space is {(heads, heads), (heads, tails), (tails, heads), (tails, tails)} and the wager has one chance in four of paying off.

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