Question

determine the missing values (indicated with ?) so that events a and b will be independent.
a) ( p(a) = 0.55 ) ( p(b) = ? ) ( p(a cap b) = 0.1375 )

b) ( p(a) = ? ) ( p(b) = \frac{3}{10} ) ( p(a cap b) = \frac{1}{7} )

Explanation:

Step1: Recall independence formula

For independent events \( A \) and \( B \), \( P(A \cap B) = P(A) \times P(B) \).

Step2: Solve part A

Given \( P(A) = 0.55 \), \( P(A \cap B) = 0.1375 \).
Using the formula: \( 0.1375 = 0.55 \times P(B) \).
Solve for \( P(B) \): \( P(B) = \frac{0.1375}{0.55} = 0.25 \).

Step3: Solve part B

Given \( P(B) = \frac{3}{10} \), \( P(A \cap B) = \frac{1}{7} \).
Using the formula: \( \frac{1}{7} = P(A) \times \frac{3}{10} \).
Solve for \( P(A) \): \( P(A) = \frac{1}{7} \div \frac{3}{10} = \frac{1}{7} \times \frac{10}{3} = \frac{10}{21} \approx 0.4762 \).

Answer:

A) \( 0.25 \)
B) \( \frac{10}{21} \) (or approximately \( 0.476 \))