Question

determining individual probabilities
consider a situation where ( p(a)=\frac{4}{5} ) and ( p(a \text { and } b)=\frac{1}{2} ).
if the events are independent, then what is ( p(b) )?
( \frac{3}{10} )
( \frac{1}{2} )
( \frac{5}{8} )
( \frac{4}{5} )

Explanation:

Step1: Use the formula for independent events

For independent events \(A\) and \(B\), \(P(A\cap B)=P(A)\times P(B)\).
Given \(P(A)=\frac{4}{5}\) and \(P(A\cap B)=\frac{1}{2}\), we substitute these values into the formula: \(\frac{1}{2}=\frac{4}{5}\times P(B)\).

Step2: Solve for \(P(B)\)

To find \(P(B)\), we use the equation \(P(B)=\frac{P(A\cap B)}{P(A)}\).
Substitute the values: \(P(B)=\frac{\frac{1}{2}}{\frac{4}{5}}\).
Using the rule for dividing fractions \(\frac{a}{b}\div\frac{c}{d}=\frac{a\times d}{b\times c}\), we have \(P(B)=\frac{1\times5}{2\times4}=\frac{5}{8}\).

Answer:

\(\frac{5}{8}\) (the third option)