Question:

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

Updated On: Apr 20, 2024
  • 514\frac{5}{14}
  • 914\frac{9}{14}
  • 815\frac{8}{15}
  • 715\frac{7}{15}
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The Correct Option is C

Solution and Explanation

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

Since A and B are independent events, we know that P(AB)=P(A)+P(B)P(AB).P(A ∪ B) = P(A) + P(B) - P(A ∩ B).
Substituting the given values, we have: 
0.65=0.25+xP(AB)0.65 = 0.25 + x - P(A ∩ B)
Since A and B are independent, the probability of their intersection (P(AB))(P(A ∩ B)) is simply the product of their individual probabilities: 
P(AB)=P(A)P(B)P(A ∩ B) = P(A) * P(B)
Substituting the values again, we have: 
0.65=0.25+x0.25x0.65 = 0.25 + x - 0.25x
Simplifying the equation, we get: 
0.65=0.25+x0.25x0.65 = 0.25 + x - 0.25x
0.650.25=0.75x0.65 - 0.25 = 0.75x
0.4=0.75x 0.4 = 0.75x
x=0.40.75x = \frac{0.4}{0.75} 
x=815x = \frac{8}{15}
Therefore, the value of x is815\frac{8}{15} . The correct option is (C) 815.\frac{8}{15} .

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