JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · Probability XII

A ship is fitted with three engines

E_{1}, E_{2}

and

E_{3}

. The engines function independently of each other with respective probabilities

\frac{1}{2}, \frac{1}{4}

and

\frac{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

X_{1}, X_{2}

and

X_{3}

denote respectively the events that the engines

E_{1}, E_{2}

and

E_{3}

are functioning. Which of the following is (are) true ?

(A) $P [X_{1}^{c} ∣ X] = \frac{3}{16}$
(B)P [Exactly two engines of the ship are functioning $∣ X] = \frac{7}{8}$ ✓ Correct
(C) $P [X ∣ X_{2}] = \frac{5}{16}$
(D) $P [X ∣ X_{1}] = \frac{7}{16}$ ✓ Correct

Answer: (B), (D)

Solution

P (X_{1}) = \frac{1}{2}

P (X_{2}) = \frac{1}{4}

P (X_{3}) = \frac{1}{4}

. The ship is operational if at least two engines function:

P (X) = P (X_{1} X_{2} X_{3}^{c}) + P (X_{1} X_{2}^{c} X_{3}) + P (X_{1}^{c} X_{2} X_{3}) + P (X_{1} X_{2} X_{3}) = \frac{1}{16} + \frac{1}{16} + \frac{1}{32} + \frac{1}{32} = \frac{1}{4}

.

(A)

P (X_{1}^{c} ∣ X) = \frac{P ( X \cap X _{1}^{c} )}{P ( X )} = \frac{1/32}{1/4} = \frac{1}{8}

✗

(B)

P (exactly 2 ∣ X) = \frac{7/32}{1/4} = \frac{7}{8}

✓

(C)

P (X ∣ X_{2}) = \frac{P ( X \cap X _{2} )}{P ( X _{2} )} = \frac{5/32}{1/4} = \frac{5}{8}

✗

(D)

P (X ∣ X_{1}) = \frac{P ( X \cap X _{1} )}{P ( X _{1} )} = \frac{7/32}{1/2} = \frac{7}{16}

✓

JEE Advanced · 2012 · 08 Apr 2012 · Paper 1 · One or more correct options