Question

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 $p(b)$?\
$\bigcirc \\ \frac{3}{10}$\
$\bigcirc \\ \frac{1}{2}$\
$\bigcirc \\ \frac{5}{8}$\
$\bigcirc \\ \frac{4}{5}$

Explanation:

Step1: Recall the formula for independent events

For independent events \( A \) and \( B \), the probability of both \( A \) and \( B \) occurring is given by \( P(A \text{ and } B)=P(A)\times P(B) \).

Step2: Substitute the given values into the formula

We know that \( P(A)=\frac{4}{5} \) and \( P(A \text{ and } B)=\frac{1}{2} \). Substituting these into the formula \( P(A \text{ and } B)=P(A)\times P(B) \), we get \( \frac{1}{2}=\frac{4}{5}\times P(B) \).

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

To solve for \( P(B) \), we can rearrange the equation. Divide both sides of the equation \( \frac{1}{2}=\frac{4}{5}\times P(B) \) by \( \frac{4}{5} \). Remember that dividing by a fraction is the same as multiplying by its reciprocal. So, \( P(B)=\frac{1}{2}\div\frac{4}{5}=\frac{1}{2}\times\frac{5}{4}=\frac{5}{8} \).

Answer:

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