Question

A ship is fitted with three engines $E_1,E_2,\text{and}{E}_3$. The engines function independently of each other with respective probabilities $\frac{1}{2},\frac{1}{4}\text{and}\frac{1}{4}.$ 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 $X_1,X_2\text{and}{X}_3$ denote respectively the events that the engines $E_1,E_2\text{and}{E}_3$ are functioning. Which of the following is (are) true?

• A. $P\left[X_1^c|X\right]=\frac{3}{16}$
• B. P[Exactly two engines of the ship are functioning |X] $=\frac{7}{8}$
• C. $P\left[X|X_2\right]=\frac{5}{16}$
• D. $P\left[X|X_1\right]=\frac{7}{16}$

Answer 1

Answer: (B), (D)

$P\left[X|X_1\right]=\frac{7}{16}$

(a) $P\left[X_1^c|X\right]=\frac{P({\overline{X}}_1\cap X)}{P\left(X\right)}$

$=\frac{\left(\frac{1}{2}\right)\left(\frac{1}{4}\right)\left(\frac{1}{4}\right)}{\frac{1}{2}.\frac{1}{4}.\frac{1}{4}+\frac{1}{2}.\frac{1}{4}.\frac{1}{4}+\frac{1}{2}.\frac{3}{4}.\frac{1}{4}+\frac{1}{2}.\frac{1}{4}.\frac{3}{4}}=\frac{1}{8}$

(b) Required probability $=\frac{\frac{3}{\mathrm{2.4.4}}+\frac{3}{\mathrm{2.4.4}}+\frac{1}{\mathrm{2.4.4}}}{\frac{1}{2}.\frac{1}{4}.\frac{3}{4}+\frac{1}{2}.\frac{3}{4}.\frac{1}{4}+\frac{1}{2}.\frac{1}{4}.\frac{1}{4}+\frac{1}{2}.\frac{1}{4}.\frac{1}{4}}=\frac{7}{8}$

(c) $P\left[X|X_2\right]=\frac{P(X\cap X_2)}{P\left(X_2\right)}$

$=\frac{\left(\frac{1}{4}\right)[\frac{1}{2}.\frac{1}{4}+\frac{1}{2}.\frac{3}{4}+\left(\frac{1}{2}\right)\left(\frac{1}{4}\right)]}{\left(\frac{1}{4}\right)}$

$=\frac{\frac{1}{4}.\frac{5}{8}}{\frac{1}{4}}=\frac{5}{8}$

(d) $\frac{P(X\cap X_1)}{P\left(X_1\right)}=\frac{\left(\frac{1}{2}\right)[\frac{1}{4}.\frac{1}{4}+\frac{3}{4}.\frac{1}{4}+\frac{1}{4}.\frac{3}{4}]}{\frac{1}{2}}=\frac{7}{16}$