Mutual Independance and Pairwise Independance

Q.

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

1. $P\left(X_1^C|X\right)=\frac{3}{16}$

2. P (Exactly two engines of the ship are functioning | X) $=\frac{7}{8}$

3. $P\left(X|X_2\right)=\frac{5}{16}$

4. $P\left(X|X_1\right)=\frac{7}{16}$

Q.

Football teams $T_1$ and $T_2$ have to play two games against each other. It is assumed that the outcomes of the two games are independent. The probabilities of $T_1$ winning, drawing and losing a game against $T_2$ are $\frac{1}{2},\frac{1}{6}$ and $\frac{1}{3}$, respectively. Each team gets 3 points for a win, 1 point for a draw and 0 points for a loss in a game. Let $X$ and $Y$ denote the total points scored by teams $T_1$ and $T_2$, respectively, after two games.

$P(X>Y)$ is

1. $\frac{1}{4}$

2. $\frac{5}{12}$

3. $\frac{1}{2}$

4. $\frac{7}{12}$

Q.

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

1. $P\left(X_1^C|X\right)=\frac{3}{16}$

2. P (Exactly two engines of the ship are functioning | X) $=\frac{7}{8}$

3. $P\left(X|X_2\right)=\frac{5}{16}$

4. $P\left(X|X_1\right)=\frac{7}{16}$