KCET · Maths · Probability
If \(A\) and \(B\) are two independent events such that \(P(\bar{A})=0.75, P(A \cup B)=0.65\) and \(P(B)=x\), then find the value of \(x\).
- A \(\frac{5}{14}\)
- B \(\frac{8}{15}\)
- C \(\frac{9}{14}\)
- D \(\frac{7}{15}\)
Answer & Solution
Correct Answer
(B) \(\frac{8}{15}\)
Step-by-step Solution
Detailed explanation
Given, \(P(\bar{A})=0.75, P(A \cup B)=0.65\) and \(P(B)=x\) \(P(A)=1-P(\bar{A})=1-0.75=0.25\)
Also, \(A\) and \(B\) are independent.
\[
\begin{aligned}
& \Rightarrow P(A \cap B)=P(A) \cdot P(B)=0.25 x \\
& \text { Using } P(A \cup B)=P(A)+P(B)-P(A \cap B), \\
& \Rightarrow \quad 0.65=0.25+x-0.25 x \Rightarrow 0.40=0.75 x \\
& \Rightarrow \quad x=\frac{40}{75}=\frac{8}{15}
\end{aligned}
\]
Also, \(A\) and \(B\) are independent.
\[
\begin{aligned}
& \Rightarrow P(A \cap B)=P(A) \cdot P(B)=0.25 x \\
& \text { Using } P(A \cup B)=P(A)+P(B)-P(A \cap B), \\
& \Rightarrow \quad 0.65=0.25+x-0.25 x \Rightarrow 0.40=0.75 x \\
& \Rightarrow \quad x=\frac{40}{75}=\frac{8}{15}
\end{aligned}
\]
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