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Intersection of Events and the Multiplication Rule

Statistics • Probability

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Intersection of Events & the Multiplication Rule

Compute the joint probability P(A ∩ B) using a 2×2 table, a two-step (tree) process, or directly from probabilities and conditional probabilities.

Enter the classification counts, then choose which row is event A and which column is event B.

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Frequently Asked Questions

What is the intersection of two events in probability?

The intersection A ∩ B means both events occur. Its probability is called the joint probability and is written as P(A ∩ B) or P(A and B).

How do you use the multiplication rule to find P(A and B)?

Use P(A ∩ B) = P(A) x P(B|A). Equivalently, you can use P(A ∩ B) = P(B) x P(A|B) if that conditional probability is known.

How can I compute conditional probability from the joint probability?

If P(A) is not 0, then P(B|A) = P(A ∩ B) / P(A). If P(B) is not 0, then P(A|B) = P(A ∩ B) / P(B).

What changes when events are independent?

If events are independent, then P(B|A) = P(B), so the multiplication rule simplifies to P(A ∩ B) = P(A) x P(B). If conditioning changes the probability, the events are dependent.

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