IIT-JEE 2012 · Paper 1 · Mathematics Q53

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MathematicsMultiple correct+4Question 53 of 60

A ship is fitted with three engines $E_1$ , $E_2$ and $E_3$ . The engines function independently of each other with respective probabilities $\frac12$ , $\frac14$ and $\frac14$ . 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?

P[X_1^c\mid X]=\dfrac3{16}

P

[Exactly two engines of the ship are functioning

\mid X

]

=\dfrac78

P[X\mid X_2]=\dfrac5{16}

P[X\mid X_1]=\dfrac7{16}

Answer and worked solution

Official answer: B, D

Let $p_1=\frac12$ , $p_2=p_3=\frac14$ .

Operational.

Exactly two: $p_1p_2q_3+p_1q_2p_3+q_1p_2p_3=\frac3{32}+\frac3{32}+\frac1{32}=\frac7{32}$ .
All three: $\frac1{32}$ .
$P(X)=\frac8{32}=\frac14$ .
(A) $P(X_1^c\cap X)=q_1p_2p_3=\frac1{32}$ , so $P(X_1^c\mid X)=\frac18$ . False.
(B) $\frac{7/32}{8/32}=\frac78$ . True.
(C) Given $E_2$ works, at least one of $E_1,E_3$ must: $1-\frac12\cdot\frac34=\frac58$ . False.
(D) Given $E_1$ works, at least one of $E_2,E_3$ : $1-\left(\frac34\right)^2=\frac7{16}$ . True.

(B), (D)

Topics:Conditional probability Independent events

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IIT-JEE 2012 · Paper 1, Mathematics question 53