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JEE Advanced2012MathematicsProbabilityActual

A ship is fitted with three engines E₁, E₂ and E₃ . The engines function independently of each other with respective probabilities 1 2 , 1 4 and 1 4 . For the ship to be operational at least two of its engines must function. Let X denote the event that the ship is operational and let X₁, X₂ and X₃ denote respectively the events that the engines E₁, E₂ and E₃ are functioning. Which of the following is(are) true?

Options

  1. AP [X₁^ c X ]= 3 16
  2. BP [Exactly two engines of the ship are functioning [X]= 7 8
  3. CP [X X₂ ]= 5 16
  4. DP [X X₁ ]= 7 16

Correct answer

B. P [Exactly two engines of the ship are functioning [X]= 7 8

Step-by-step solution

Given that P (X₁ )= 1 2 , P (X₂ )= 1 4 , P (X₃ )= 1 4 P(X)=P (at least 2 engines are functioning) aligned =P (X₁ X₂ .& . X₃^ C )+P (X₁ X₂^ C X₃ ) &+P (X₁^ C X₂ X₃ )+P (X₁ X₂ X₃ ) aligned = 1 2 1 4 3 4 + 1 2 3 4 1 4 + 1 2 1 4 1 4 + 1 2 1 4 1 4 = 1 4 (a) P (X₁^ C / X )= P (X₁^ C X ) P(X) = P (X₁^ C X₂ X₃ ) P(X) = 1 2 1 4 1 4 1 4 = 1 8 (a) is not true. (b) P [Exactly two engines are functioning / X]= P[( Exactly two engines are functioning ) X] P(X) = P (X₁^ C X₂ X₃ )+P (X₁ X₂^ C X₃ )+P (X₁ X₂ X₃^ C ) P(X) = 1 2 1 4 1

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