Question

two events, a and b, are independent of each other. $p(a)=\frac{1}{6}$ and $p(a \text{ and } b)=\frac{1}{8}$. what is $p(b)$ written as a decimal? round to the nearest hundredth, if necessary.
\bigcirc 0.02
\bigcirc 0.04
\bigcirc 0.29
\bigcirc 0.75

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: Solve for \( P(B) \)

We know \( P(A) = \frac{1}{6} \) and \( P(A \text{ and } B) = \frac{1}{8} \). Rearranging the formula to solve for \( P(B) \), we get \( P(B) = \frac{P(A \text{ and } B)}{P(A)} \).

Substitute the given values: \( P(B) = \frac{\frac{1}{8}}{\frac{1}{6}} \).

Dividing by a fraction is the same as multiplying by its reciprocal, so \( P(B) = \frac{1}{8} \times \frac{6}{1} = \frac{6}{8} = \frac{3}{4} \).

Convert \( \frac{3}{4} \) to a decimal: \( \frac{3}{4} = 0.75 \).

Answer:

0.75