Question

A ship is fitted with three engines $E_1,E_2$ and $E_3$. 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 $X_1,X_2$ 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. $P\left[X_1^c|X\right]=3/16$
• B. $P\left[\text{Exactly two engines of the ship are functioning}|X\right]=7/8$
• C. $P\left[X|X_2\right]=5/16$
• D. $P\left[X|X_1\right]=7/16$

Answer 1

Answer: (B), (D)

$P\left[X|X_1\right]=7/16$
$P\left[\text{Exactly two engines of the ship are functioning}|X\right]=7/8$

$P\left(X_1\right)=\frac{1}{2},P\left(X_2\right)=\frac{1}{4},P\left(X_3\right)=\frac{1}{4}$
$P\left(X\right)=P(X_1\cap X_2\cap X_3^c)+P(X_1\cap X_2^c\cap X_3)+P(X_1^c\cap X_2\cap X_3)+P(X_1\cap X_2\cap X_3)=\frac{1}{4}$

(A) $P\left(X_1^c/X\right)=\frac{P(X\cap X_1^c)}{P\left(X\right)}=\frac{\frac{1}{2}\times \frac{1}{4}\times \frac{1}{4}}{\frac{1}{4}}$
$=\frac{1}{8}$
$P\left(\frac{\text{exactly 2 engines}}{X}\right)=\frac{(\frac{1}{2}\times \frac{1}{4}\times \frac{1}{4})+(\frac{1}{2}\times \frac{1}{4}\times \frac{3}{4})+(\frac{1}{2}\times \frac{3}{4}\times \frac{1}{4})}{\frac{1}{4}}$
$=\frac{7}{8}$
$P\left(\frac{X}{X_2}\right)=\frac{P(X\cap X_2)}{P\left(X_2\right)}=\frac{(\frac{1}{2}\times \frac{1}{4}\times \frac{1}{4})+(\frac{1}{2}\times \frac{1}{4}\times \frac{3}{4})+(\frac{1}{2}\times \frac{1}{2}\times \frac{1}{4})}{\frac{1}{4}}$
$=\frac{\frac{5}{32}}{\frac{1}{4}}$

$=\frac{5}{8}$
$P\left(\frac{X}{X_1}\right)=\frac{P(X\cap X_1}{P\left(X_1\right)}=\frac{\frac{7}{32}}{\frac{1}{2}}$
$=\frac{7}{16}$