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JEE Advance - Mathematics (2012 - Paper 1 Offline - No. 3)

A ship is fitted with three engines $${E_1},{E_2}$$ and $${E_3}$$. The engines function independently of each other with respective probabilities $${1 \over 2},{1 \over 4}$$ and $${1 \over 4}$$. 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?
$$P\left[ {X_1^c|X} \right] = {3 \over {16}}$$
$$P$$ [exactly two engines of the ship are functioning $$\left. {|X} \right] = {7 \over 8}$$
$$P\left[ {X|{X_2}} \right] = {5 \over {16}}$$
$$P\left[ {X|{X_1}} \right] = {7 \over {16}}$$

Explanation

$$\begin{array}{r} \text { Given, } \mathrm{P}\left(\mathrm{X}_1\right)=\frac{1}{2} \Rightarrow \mathrm{P}\left(\overline{\mathrm{X}}_1\right)=1-\frac{1}{2}=\frac{1}{2} \\ \mathrm{P}\left(\mathrm{X}_2\right)=\frac{1}{4} \Rightarrow \mathrm{P}\left(\bar{X}_2\right)=1-\frac{1}{4}=\frac{3}{4} \\ \mathrm{P}\left(\mathrm{X}_3\right)=\frac{1}{4} \Rightarrow \mathrm{P}\left(\bar{X}_3\right)=1-\frac{1}{4}=\frac{3}{4} \end{array}$$

$$\begin{aligned} & \text { Now, } P(X)=P\left(E_1 E_2 E_3\right)+P\left(\bar{E}_1 E_2 E_3\right)+P\left(E_1\right. \\ & \left.\overline{\mathrm{E}}_2 \mathrm{E}_3\right)+\mathrm{P}\left(\mathrm{E}_1 \mathrm{E}_2 \overline{\mathrm{E}}_3\right) \\ & =\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{4}+\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{1}{4}+\frac{1}{2} \cdot \frac{3}{4} \cdot \frac{1}{4}+\frac{1}{2} \cdot \frac{1}{4} \cdot \frac{3}{4} \\ \end{aligned}$$

$$\begin{aligned} & P(X)=\frac{1}{32}+\frac{1}{32}+\frac{3}{32}+\frac{3}{32} \\ & P(X)=\frac{1}{4} \end{aligned}$$

(i) $$\mathrm{P}\left(\frac{\mathrm{X}_1^c}{\mathrm{X}}\right)=\frac{\mathrm{P}\left(\mathrm{X}_1^c \cap \mathrm{X}\right)}{\mathrm{P}(\mathrm{X})}$$

Now, $$X_1^c \cap X$$ indicates the event that the ship is operational but $$\mathrm{E}_1$$ is not functioning.

$$\begin{aligned} & \therefore \quad \mathrm{P}\left(\mathrm{X}_1^c \cap \mathrm{X}\right)=\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}=\frac{1}{32} \\ & \therefore \mathrm{P}\left(\frac{\mathrm{X}_1^c}{\mathrm{X}}\right)=\frac{\frac{1}{32}}{\frac{1}{4}}=\frac{1}{8} \end{aligned}$$

(ii) P (Exactly two engines of the ship are functioning $$\mid X$$)

$$\begin{aligned} & =\frac{\mathrm{P}\left(\overline{\mathrm{E}}_1 \mathrm{E}_2 \mathrm{E}_3\right)+\mathrm{P}\left(\mathrm{E}_1 \overline{\mathrm{E}}_2 \mathrm{E}_3\right)+1\left(\mathrm{E}_1 \mathrm{E}_2 \overline{\mathrm{E}}_3\right)}{\mathrm{P}(\mathrm{X})} \\ & =\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{3}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{4}} \\ & =\frac{7}{8} \end{aligned}$$

$$\begin{aligned} \text { (iii) } & P\left(X \mid X_2\right)=\frac{P\left(X \cap X_2\right)}{P\left(X_2\right)} \\ & =\frac{P\left(E_1 E_2 E_3\right)+P\left(\bar{E}_1 E_2 E_3\right)+P\left(E_1 E_2 \bar{E}_3\right)}{P\left(X_2\right)} \\ & =\frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{4}} \\ & =\frac{5}{8} \end{aligned}$$

$$\begin{aligned} \text { (iv) } & P\left(X \mid X_1\right)=\frac{P\left(X \cap X_1\right)}{P\left(X_1\right)} \\ = & \frac{P\left(E_1 E_2 E_3\right)+P\left(E_1 \bar{E}_2 E_3\right)+P\left(E_1 E_2 \bar{E}_3\right)}{P\left(X_1\right)} \\ = & \frac{\frac{1}{2} \times \frac{1}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{3}{4} \times \frac{1}{4}+\frac{1}{2} \times \frac{1}{4} \times \frac{3}{4}}{\frac{1}{2}} \\ = & \frac{7}{16} \end{aligned}$$

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