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Determine event independence using P(A and B)=P(A)P(B).

Determine whether two events are independent using \(P(A and B) = P(A)P(B)\)

Problem
Are events \(A\) and \(B\) independent? Use these probabilities: \(P(A)=1/2\), \(P(B)=1/3\), \(P(A~\text{and}~B)=1/6\).
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Hint

Multiply \(P(A)\) and \(P(B)\) and compare that product to \(P(A and B)\).

Events are independent when \(P(A and B)=P(A)P(B)\).

Solution walkthrough

01

State the independence test

\[P(A~\text{and}~B)=P(A)P(B)\]

This equality is the defining probability test when the supplied probabilities are valid.

02

Substitute the marginal probabilities

\[P(A)P(B)=(1/2)(1/3)\]

The values one-half and one-third come from the prompt.

03

Multiply

\[(1/2)(1/3)=1/6\]

Multiplying numerators and denominators gives one-sixth.

04

Compare and conclude

\[P(A~\text{and}~B)=1/6=P(A)P(B)~->~\text{independent}\]

The observed intersection probability matches the product exactly.

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Another way

  1. Check the conditional probability: P(A given B)=(1/6)/(1/3)=1/2=P(A).

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Common mistake

Do not add P(A) and P(B); independence uses the product of marginal probabilities.

Solution walkthrough video