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) )?
options: ( \frac{3}{10} ), ( \frac{1}{2} ), ( \frac{5}{8} ), ( \frac{4}{5} )

Explanation:

Step1: Recall the formula for independent events

For independent events, \( P(A \text{ and } B) = P(A) \times P(B) \)

Step2: Substitute the given values

We know \( P(A) = \frac{4}{5} \) and \( P(A \text{ and } B) = \frac{1}{2} \). Substitute into the formula:
\( \frac{1}{2} = \frac{4}{5} \times P(B) \)

Step3: Solve for \( P(B) \)

To find \( P(B) \), divide both sides by \( \frac{4}{5} \):
\( P(B) = \frac{1}{2} \div \frac{4}{5} = \frac{1}{2} \times \frac{5}{4} = \frac{5}{8} \)

Answer:

\(\frac{5}{8}\) (corresponding to the option \(\frac{5}{8}\))