A ship is fitted with three engines E1,E2and E3 . The engine function independently of each other with respective probabilities 12,14,14 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,X2and X3 denotes respectively the events that the engines E1,E2and E3 are functioning. Which of the following is (are) true?

A ship is fitted with three engines $E_{1}, E_{2} a n d E_{3}$ . The engine function independently of each other with respective probabilities $\frac{1}{2}, \frac{1}{4}, \frac{1}{4}$

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 $X_{1}, X_{2} and X_{3}$ denotes respectively the events that the engines $E_{1}, E_{2} a n d E_{3}$ are functioning. Which of the following is (are) true?

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$P [X_{1}^{c} | x] = \frac{3}{16}$

P [Exactly two engines of the ship are functioning | X] is $\frac{7}{8}$

$P [X | X_{2}] = \frac{5}{16}$

$P [X | X_{1}] = \frac{7}{16}$

answer is B, D.

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Detailed Solution

$\begin{matrix} P (E_{1}) = \frac{1}{2}, P (E_{2}) = \frac{1}{4}, P (E_{3}) = \frac{1}{4} \\ P (x) = P (E_{1} E_{2} E_{3}) + P ({\bar{E}}_{1} E_{2} E_{3}) + P (E_{1} {\bar{E}}_{2} E_{3}) + P (E_{1} E_{2} {\bar{E}}_{3}) \\ = \frac{1}{2} . \frac{1}{4} . \frac{1}{4} + \frac{1}{2} . \frac{1}{4} . \frac{1}{4} + \frac{1}{2} . \frac{3}{4} . \frac{1}{4} + \frac{1}{2} . \frac{1}{4} . \frac{3}{4} \\ = \frac{1}{4} \end{matrix}$

$P (x_{2}) = \frac{1}{4}$

$P (x_{3}) = \frac{1}{4}$

$P (x) = P (E_{1} E_{2} E_{3}) + P ({\bar{E}}_{1} E_{2} E_{3}) + P (E_{1} {\bar{E}}_{2} E_{3}) + P (E_{1} E_{2} {\bar{E}}_{3})$

$= \frac{1}{2} . \frac{1}{4} . \frac{1}{4} + \frac{1}{2} . \frac{1}{4} . \frac{1}{4} + \frac{1}{2} . \frac{3}{4} . \frac{1}{4} + \frac{1}{2} . \frac{1}{4} . \frac{3}{4}$