Question

If A and B are two independent events such that \(P(\overline{A}) = 0.75\), \(P(A \cup B) = 0.65\) and \(P(B) = x\), then find the value of x:

• A. \(\frac{5}{14}\)
• B. \(\frac{9}{14}\)
• C. \(\frac{8}{15}\)
• D. \(\frac{7}{15}\)

Answer 1

The Correct Option is C

We have the following information:
\(P(A) = 0.25 \)
\(P(A ∪ B) = 0.65 \)
\(P(B) = x \)

Since A and B are independent events, we know that \(P(A ∪ B) = P(A) + P(B) - P(A ∩ B). \)
Substituting the given values, we have: 
\(0.65 = 0.25 + x - P(A ∩ B) \)
Since A and B are independent, the probability of their intersection \((P(A ∩ B))\) is simply the product of their individual probabilities: 
\(P(A ∩ B) = P(A) * P(B) \)
Substituting the values again, we have: 
\(0.65 = 0.25 + x - 0.25x\)
Simplifying the equation, we get: 
\(0.65 = 0.25 + x - 0.25x \)
\(0.65 - 0.25 = 0.75x\)
\( 0.4 = 0.75x \)
\(x = \frac{0.4}{0.75}\) 
\(x = \frac{8}{15} \)
Therefore, the value of x is\(\frac{8}{15} \). The correct option is (C) \(\frac{8}{15} .\)