Question

if ( p(a)=\frac{1}{2}, p(a \text{ and } b)=\frac{1}{6} ), and ( p(a \text{ or } b)=\frac{2}{3} ), what is ( p(b) )?

( \frac{2}{3} )

( \frac{1}{3} )

( \frac{1}{4} )

( \frac{1}{2} )

Explanation:

Step1: Recall the formula for \(P(A\ or\ B)\)

The formula is \(P(A\ or\ B)=P(A)+P(B)-P(A\ and\ B)\).

Step2: Rearrange the formula to solve for \(P(B)\)

We get \(P(B)=P(A\ or\ B)+P(A\ and\ B)-P(A)\).

Step3: Substitute the given values

Substitute \(P(A)=\frac{1}{2}\), \(P(A\ and\ B)=\frac{1}{6}\), and \(P(A\ or\ B)=\frac{2}{3}\) into the formula:

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Answer:

\(\frac{1}{3}\) (the second option)