Conditional probability

Read the event after the vertical bar first. It tells you the new denominator.

• The bar $\mid$ is read “given”. In $P(A\mid B)$, B is the condition and becomes the denominator.
• In $P(B\mid A)$, the denominator is A.
• The condition must be possible: $P(B)>0$. The two conditional probabilities are usually different.

1. Circle the condition after the bar. It tells you which outcomes remain and supplies the denominator.
2. Keep the target inside that condition. This is the intersection in the numerator.
3. Divide target-and-condition by all-condition. Simplify only after the correct numerator and denominator are visible.

• Two-way table: target-and-condition cell divided by the condition row or column total.
• Venn diagram: overlap divided by the whole condition circle.
• Possibility grid: successful cells inside the condition cells.
• Tree: target-and-condition path probability divided by the total probability of every path satisfying the condition.

The probability test for independence is simply:

If restricting the world to B leaves the probability of A unchanged, then B has provided no information about A. That is exactly what independence means.