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:

\[ P(A ∩ B) = P(A) \cdot P(B) \]

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) \(\frac{8}{15}\).

Answer 2

Given: 
\[ P(\overline{A}) = 0.75 \Rightarrow P(A) = 1 - 0.75 = 0.25 \] \[ P(A \cup B) = 0.65,\quad P(B) = x \] Since A and B are independent, we have: \[ P(A \cap B) = P(A) \cdot P(B) = 0.25 \cdot x \] Now use the formula: \[ P(A \cup B) = P(A) + P(B) - P(A \cap B) \] Substituting: \[ 0.65 = 0.25 + x - 0.25x \] \[ 0.65 = 0.25 + x(1 - 0.25) = 0.25 + 0.75x \] \[ 0.65 - 0.25 = 0.75x \Rightarrow 0.40 = 0.75x \] \[ x = \frac{0.40}{0.75} = \frac{40}{75} = \frac{8}{15} \] Correct Answer: \(\frac{8}{15}\)