Event

Q.

The probability of a sure event is

1. $0$

2. $1$

3. $2$

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

Q.

Given two independent events A and B such that P(A) = 0.3, P(B) = 0.6 Find
P(A and B)

P(A and not B)

P(A and B)

P (neither A nor B)

Q. For a biased die, the probability of getting an even number is twice the probability of getting an odd number. The die is thrown twice and the sum of the outcomes is even. Then the probability that both the outcomes on the die is an odd number is

1. $\frac{1}{3}$
2. $\frac{3}{10}$
3. $\frac{1}{9}$
4. $\frac{1}{5}$

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. $\frac{1}{2}$

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

3. $\frac{11}{36}$

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

Q.

If A and B are two events of sample space S associated with a random experiment, then occurrence of A provided B has already occurred is called conditional probability. It is denoted by

1. P(A)/P(B)

2. P(B/A)

3. P(A)P(B)

4. P(A/B)

Q.

If three six faced die each marked with numbers 1 to 6 on six faces, the thrown find the total number of possible outcomes.

Q.

Define capacity?

Q.

A pair of dice is rolled. If the outcome is a doublet, a coin is tossed. Deterrnine the total number of elementary events associated to this experiment.

Q. Three distinguishable ball distributed in three cells. Find the conditional probability that all the three occupy the same cell, given that at least two of them are in the same cell.

Q. If a sample space has $4$ elements, then number of events associated with it is

Q.

An electronic assembly consists of two subsystems, say A and B. From previous testing procedures, the following probabilities are assumed to be known

P(A fails)= 0.2, P(B fails alone) =0.15, P(A and B fail)=0.15

Evaluate the following probabilities

P (A fails/B has failed )

P(A fails alone)

Q.

The probability of a sure event is$:$

Q. A player $X$ has a biased coin whose probability of showing heads is $p$ and a player $Y$ has a fair coin. They start playing a game with their own coins and play alternately. The player who throws a head first is a winner. If $X$ starts the game, and the probability of winning the game by both the players is equal, then the value of ${}^{\prime }{p}^{\prime }$ is :

1. $\frac{1}{3}$
2. $\frac{1}{5}$
3. $\frac{2}{5}$
4. $\frac{1}{4}$

Q.

If I have a $20$ sided dice and roll it twice, what are the odds that I will roll the same number twice?

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.

What is the total number of elementary events associated to the random experiment of throwing three dice together?

Q.

List all events associated with the andom experiment of tossing of two coins. How many of them arc elementary events ?

Q. There are $5$ multiple choice questions (only one correct option) in a test. If the first three questions have $4$ choices each and the next two have $5$ choices each, then number of possible ways in which a student can answers all the question is

1. $1500$
2. $1600$
3. $1700$
4. $1800$

Q.

Two dice are thrown simultaneously. If X denotes the number of sixes, then the expectation of X is:

1. 1/3

2. 1/6

3. 1/2

4. 1/4

Q.

What is the meaning of fair trial and how does inclusion of fair trial influence the result?

Q. Define event

Q.

A coin is tossed. Find the total number of elementary events and also the total number events associated with the random experiment.

Q. When a fair die is thrown twice, let $(a,b)$ denote the outcome in which the first throw shows $a$ and the second throw shows $b.$ Consider the following events: $A=\left\{\right(a,b\left)|a\text{is odd}\right\},B=\left\{\right(a,b\left)|b\text{is odd}\right\}$ and $C=\left\{\right(a,b)|a+b\text{is odd}\},$ then which of the following(s) is(are) correct?

1. $P(A\cap B)=P(B\cap C)$
2. $P(A\cap C)=P(B\cap C)$
3. $P(A\cap B\cap C)=0$
4. $P\left(A\right)=P\left(B\right)=P\left(C\right)$

Q. The probability of an impossible event is

1. $1$
2. $0$
3. less than $0$
4. greater than $1$

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. A locker can be opened by dialing a fixed three-digit code (between $000$ and $999$). A stranger, who does not know the code, tries to open the locker by dialing three digits at random. If $p$ is the probability that the stranger succeeds at the $kth$ trial, then the value of $1000p$ is

Q.

Probability fo solving specific problem independently by A and B are $\frac{1}{2}and\frac{1}{3}$ respectively. If both try to solve the problem independently, Find the probability that
the problem is solved

Probability fo solving specific problem independently by A and B are $\frac{1}{2}and\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that
Exactly one of them solves the problem

One card is drawn at random from a well- shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
E: the card drawn is a spade, F: the card drawn is an ace

Q. The numbers a, b, c r selected by throwing a dice thrice then the probability that
a b c are in AP is

Q. Four dice (six faced) are rolled. The number of possible outcomes in which at least one die shows 2 is

1. 1296
2. 625
3. 671
4. None of these

Q. Twelve balls are distributed among three boxes. The probability that the first box will contains three balls.

1. $\frac{2^9}{3^{12}}$
2. $\frac{{}^{12}{C}_3\times 2^9}{3^{12}}$
3. $\frac{{}^{12}{C}_3\times 2^9}{{12}^3}$
4. $\frac{{}^{12}{C}_3}{3^{12}}$