Event

Q.

Box I contain three cards bearing numbers 1, 2, 3; box II contains five cards bearing numbers 1, 2, 3, 4, 5; and box III contains seven cards bearing numbers 1, 2, 3, 4, 5, 6, 7. A card is drawn from each of the boxes. Let $x_i$ be the number on the card drawn from the $i^{th}$ box $i=1,2,3.$

The probability that $x_1,x_2$ and $x_3$ are in arithmetic progression, is ?

1. $\frac{9}{105}$

2. $\frac{10}{105}$

3. $\frac{11}{105}$

4. $\frac{7}{105}$

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 $0$ point 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. $\frac{11}{36}$
2. $\frac{1}{3}$
3. $\frac{13}{36}$
4. $\frac{1}{2}$

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. $\frac{11}{36}$

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

3. $\frac{13}{36}$

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

Q. The letters of the word $\text{PROBABILITY}$ are written down at random in a row. Let $E_1$ denote the event that two $I^{\prime }s$ are together and $E_2$ denote the event that two $B^{\prime }s$ are together. Then which of the following is (are) CORRECT?

1. $P\left(E_1\right)=P\left(E_2\right)=\frac{2}{11}$
2. $P(E_1\cap E_2)=\frac{2}{55}$
3. $P(E_1\cup E_2)=\frac{18}{55}$
4. $P\left(E_1|E_2\right)=\frac{1}{5}$

Q. Three numbers are chosen at random without replacement from {1, 2, ......, 15}. Let $E_1$ be the event that minimum of the chosen numbers is 5 and $E_2$ be that their maximum is 10 then:

1. $P\left(E_1\right)=\frac{9}{91}$
2. $P\left(E_2\right)=\frac{36}{455}$
3. $P(E_1\cap E_2)=\frac{4}{455}$
4. $P\left(E_1/E_2\right)=\frac{1}{9}$

Q.

While throwing a pair of dice, an event A is defined as 'sum of faces will be at least 10'. Find the total number of favorable outcomes.

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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. $\frac{11}{36}$

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

3. $\frac{13}{36}$

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