$A$ and $B$ are two events of a random experiment such that $P(B)=0.4, P(A \cap \bar{B})=0.5, P(A \cup B)+P\left(\frac{B}{A \cup \bar{B}}\right)=1.15$ then $P(A)=$

There are two boxes each containing 10 balls. In each box, few of them are black balls and rest are white. A ball is drawn at random from one of the boxes and found that it is black. If the probability that the black ball drawn is from the second box is $\frac{1}{5}$, then number of black balls in the first box is

In a shelf there are three mathematics and two physics books. A student takes a book randomly. If he randomly takes, successively for three time by replacing the book already taken every time, then the mean of the number of mathematics books which is treated as random variable is

In possion distribution, if $\frac{P(x=5)}{P(X=2)}=\frac{1}{7500}$ and $\frac{P(X=5)}{P(X=3)}=\frac{1}{500}$, then the mean of the distribution is

If two smallest squares are chosen at random on a chess board, then the probability of getting these squares such that they do not have a side in common is

Let $A$ and $B$ be two events in a random experiment . If $P(A \cap \bar{B})=0.1, P(\bar{A} \cap B)=0.2$ and $P(B)=0.5$, then $P(\bar{A} \cap \bar{B})=$

An urn contains 7 red, 5 white and 3 black balls. Three balls are drawn randomly one after the other without replacement. If it is known that first ball drawn is red and the second ball drawn is white, then the probability that the third ball drawn is not red is

The range of a discrete random variable $X$ is $\{1,2,3\}$ and the probabilities of its elements are given by $P(X=1)=3 k^3, P(X=2)=2 k^2$ and $P(X=3)=7-19 \mathrm{k}$. Then, $P(X=3)=$

Among every 8 units of a product, one is likely to be defective. If a consumer has order 5 units of that product, then the probability that atmost one unit is defective among them is

Out of the given 25 consecutive position integers, three integers are drawn. If the least integer among given 25 integers is an odd number, then the probability that the sum of the three integers drawn is an even number is

If three dice are thrown at a time, then the probability of getting the sum of the numbers on them as a prime number is

Three companies $C_1, C_2, C_3$ produce car tyres. A car manufacturing company buys $40 \%$ of its requirement from $C_1, 35 \%$ from $C_2$ and $25 \%$ from $C_3$. The company knows that $2 \%$ of the tyres supplied by $C_1, 3 \%$ by $C_2$ and $4 \%$ by $C_3$ are defective. If a tyre chosen random from the consignment received is found defective then, the probability that it was supplied by $C_2$ is

If the mean and variance of a binomial distribution are $\frac{4}{3}$ and $\frac{10}{9}$ respectively, then $P(X \geq 6)=$

If a number $x$ is drawn randomly from the set of numbers $\{1,2,3, \ldots ., 50\}$, then the probability that number $x$ that is drawn satisfies the inequation $x+\frac{10}{x} \leq 11$ is

If a coin is tossed seven times, then the probability of getting exactly three heads such that number two heads occur consecutively is

Two cards are drawn randomly from a pack of 52 playing cards one after the other with replacement. If $A$ is the event of drawing a face card in first draw and $B$ is the event of drawing a clubs card in second draw, then $P\left(\frac{\bar{B}}{A}\right)=$

If $X$ is a random variable with probability distribution $P(X=k)=\frac{(2 k+3) c}{3^k}, k=0,1,2, \ldots .$. to $\infty$, then $P(X=3)=$

Let $P=\left[\begin{array}{lll}1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9\end{array}\right]$ be a matrix. Three elements of this matrix $P$ are selected at random. $A$ is the event of having the three elements whose sum is odd. $B$ is the event of selecting the three elements which are in a row or column. Then, $P(A)+P\left(\frac{A}{B}\right)=$

$A, B_1, B_2, B_3$ are the events in a random experiment. If $P\left(B_1\right)=0.25, P\left(B_2\right)=0.30, P\left(B_3\right)=0.45, P\left(\frac{A}{B_1}\right)=0.05$, $P\left(\frac{A}{B_2}\right)=0.04, P\left(\frac{A}{B_3}\right)=0.03$, then $P\left(\frac{B_2}{A}\right)=$

$A, B$ are the events in a random experiment.

If $P(A)=\frac{1}{2}, P(B)=\frac{1}{3}, P(A \cap B)=\frac{1}{4}$, then $P\left(\frac{A^c}{B^c}\right)+P\left(\frac{A}{B}\right)=$

Two persons $A$ and $B$ play a game by throwing two dice. If the sum of the numbers appeared on the two dice is even, A will get $\frac{1}{2}$ point and $B$ will get $\frac{1}{2}$ point.

If the sum is odd, A will get one point and $B$ will get no point. The arithmetic mean of the random variable of the number of points of $A$ is

If three smallest squares are chosen at-random on a chess board, then the probability of getting them in such a way that they are all together in a row or in a column is

If three cards are drawn randomly from a pack of 52 playing cards then the probability of getting exactly, one spade card, exactly one king and exactly one card having a prime number is

Urn A contains 6 white and 2 black balls; run B contains 5 white and 3 black balls and urn C contains 4 white and 4 black balls. if an urn is chosen at random and a ball is drawn at random from it, then the probability that the ball drawn is white is

If a random variable $X$ has the following probability distribution, then its variance is

If a matrix is chosen at random from the set of all $3 \times 3$ non-zero matrices whose entries are the elements of the set $\{-1,0,1\}$, then the probability that the matrix is skew-symmetric is

A boy throws an unbiased die. Whenever he gets 1 on the die he has a further chance to throw it once again immediately. The probability that the boy gets a score of 7 in this process is

There are 10 coins in a box out of which 8 are normal and the remaining are with heads on both sides. A coin is chosen at random from the box and tossed 6 times. If it shows heads each time, then the probability that the selected coin has head on both sides is

$ \text { A random variable } X \text { has the following distribution, } $

$ \begin{array}{lllllll} \hline X=x_i & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P\left(X=x_i\right) & 0.1 & k & 0.2 & 2 k & 3 k & k \\ \hline \end{array} $

Then, the variance of this distribution is

A bag contains four balls. Two balls are drawn randomly and found them to be white. The probability that all the balls in the bag are white is

If the coefficients $a$ and $b$ of a quadratic expression $x^2+a x+b$ are chosen from the sets $A=\{3,4,5\}$ and $B=\{1,2,3,4\}$ respectively, then the probability that the equation $x^2+a x+b=0$ has real roots is

A random variable $X$ has the following probability distribution

$ \begin{array}{|c|l|l|l|l|l|l|l|l|} \hline \boldsymbol{X}=\boldsymbol{x} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ \hline \boldsymbol{P}(\boldsymbol{X}=\boldsymbol{x}) & 0.15 & 0.23 & k & 0.10 & 0.20 & 0.08 & 0.07 & 0.05 \\ \hline \end{array} $

For the events $E=\{x / x$ is a prime number $\}$ and $F=\{x / x<4\}$, then $P(E \cup F)=$

5 persons entered a lift cabin in the cellar of a 7 floor building apart from cellar. If each of them independently and with equal probability can leave the cabin at any floor out of the 7 floors beginning with the first, then the probability of all the 5 persons leaving the cabin at different floors is

A bag contains 3 red, 5 black and 7 blue balls. If three balls are drawn at random simultaneously from the bag, then the probability of getting at least two blue balls is

In a game, two dice are thrown simultaneously by a person $A$ and two cards are drawn at random simultaneously from a pack of 52 playing cards by a person $B$. They win the game, if $A$ gets a prime score as the sum of the numbers appear on both the dice and $B$ gets a face card and a card having a prime number. Then, the probability that both $A$ and $B$ win is

Two players $A$ and $B$ alternatively toss 3 coins simultaneously. The player who gets 2 heads and 1 tail first, wins the game. If game continues until someone wins and if $A$ begins the game, the probability that B wins the game is

If two cards are drawn at random simultaneously from a pack of 52 playing cards, then the probability of getting a face card and a spade card other than the face card is

If three unbiased dice are rolled simultaneously, then the probability that all the three dice show distinct numbers is

Three persons $A, B$ and $C$ attended a recruitment test, The ratio of the chances of $A, B, C$ in getting through the test is $1: 2: 3$ and their probabilities to face the interview successfully are $0.8,0.7,0.6$, respectively. If one of them is to be selected for the post, then the probability that $A$ gets the post is

Two cards are drawn at random one after the other with replacement from a pack of 52 playing cards. Then, the variance of the random variable of the number of spade cards among the drawn cards is

If $A$ and $B$ are two events of a random experiment such that $P(A \cup B)=P(A \cap B)$, then which one amongst the following four options is not true?

If a group of six students including two particular students $A$ and $B$ stand in a row, then the probability of getting an arrangement in which $A$ and $B$ are separated by exactly one student in between them is

$A, B, C, D$ cut a pack of 52 well shuffled playing cards successively in the same order. If the person who cuts a spade first, wins the game and the game continues until this happens, then the probability that $A$ wins the game is

Two bad eggs are mixed accidentally with 10 good ones. If three eggs are drawn at random from this lot in succession without replacement, then the variance of the probability distribution of the number of bad eggs drawn is

The probability of getting a king and a spade card when two cards are drawn simultaneously from a pack of 52 playing card is

Two cards are drawn from a pack of 52 playing cards one after the other. If $p_1$ is the probability of getting a queen in the first draw and a diamond card in the second draw when the first card drawn is replaced and $p_2$ is the probability of the same event when the first card drawn is not replaced. Then $\frac{p_1}{p_2}=$

Bag $A$ contains 4 white and 2 black balls, bag $B$ contains 3 white and 3 black balls and bag $C$ contains 2 white and 4 black balls. If a bag is chosen at random and a ball is chosen at random from it, then the probability that the ball drawn is black is

A random variable $X$ has the following probability distribution

$ \begin{array}{llllllllll} \hline X=\mathbf{x}_i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\ \hline P\left(X=\mathbf{x}_i\right) & 10 k & 9 k & 8 k & 8 k & 6 k & 5 k & 4 k & 3 k & k \\ \hline \end{array} $

where $k$ is a real number.

If $A=\left\{x_i \mid x_i\right.$ is a prime number $\}$ and $B=\left\{x_i \mid x_i>5\right\}$ are two events, then $P(A \cup B)=$

If $X$ is a Poisson variate such that $\frac{5}{3} k=P(X=2) =P(X=3)$, then $P(X=5)=$

A bag contains 9 identical black balls numbered 1 to 9 . and 4 identical white balls numbered 1 to 4 . If 3 balls are drawn at a time randomly from that bag, then the probability of getting atleast one white ball is

The probabilities of two persons to hit a target are $1 / 4$ and $1 / 5$ respectively. The probability that the target is being hit when both of them attempt independently is

When 3 dice are thrown at a time, the sum of the numbers appeared on 3 dice were found to be 15 . Then, the probability that the number 5 does not appear on any one of the dice is

If the probability distribution of a random variable $X$ is given by

$ \begin{array}{|c|c|c|c|c|c|c|} \hline X=x & 0 & 2 & 4 & 6 & 8 & 10 \\ \hline P(X=x) & 0 & k & 2 k & 5 k^2 & 2 k^2 & 3 k \\ \hline \end{array} $

then the mean of $X$ is

The probability of getting a success in a trail is five times that of a failure. The probability of getting atmost one success in 5 trails, is

A bag contains 3 white and 6 red balls. Four balls are drawn at a time randomly. Then, the probability of getting at least two red balls is

$A$ and $B$ are two independent events $P(A)=\frac{2}{5}, P(B)=\frac{1}{3}$.

Match the following :

The correct match is

Two players $A$ and $B$ are alternately throwing a coin and a die together. $A$ player who first throws head and 6 wins the game. If $A$ starts the game, then the probability that $B$ wins the game is

If two dice are thrown and if $X$ denotes the sum of the numbers that show up on the faces of the dice, then the mean of the random variable $X$ is

In a university campus, the probability that a person chosen at random is an engineering student is $\frac{1}{5}$. The probability of having atmost two engineering students in a sample of 8 people is

When two dice are thrown, the probability of getting an ordered pair $(x, y)$ such that $x^2+y^2 \leq 25$ where $x$ and $y$ are numbers that show up on the two dice, is

If two cards are drawn simultaneously from a well shuffled pack of 52 cards, then the probability of getting a card having a prime number and a card having a number which is a multiple of 5 is

If $A$ and $B$ are two events of a random experiment such that $P(\bar{A})=\frac{2}{3}, P(B)=\frac{4}{15}$ and $P(A \cap \bar{B})=\frac{1}{5}$, then $\sqrt{195[P(B \mid(A \cup \bar{B}))+P(A \cup B)]}=$

A random variable $X$ has the range $\{0,1,2, \ldots$.$\} . If P(X=r)=k(1+r) 3^{-r}$, for $r=0,1,2, \ldots$ where $k>0$ is a real number, then $P(X=0)+P(X=1)+P(X=2)=$

In an experiment a person gets success $\alpha$ times out of $\beta$ trails. If the experiment consists of $n$ trials, then the probability that he fails at least $(n-1)$ times is

When two dice are thrown, the probability of getting a prime number on die and a composite number on the other is

Let $A, B, C$ be three pairwise independent events of a random experiment. If $P(\bar{B} \cup \bar{C})=\frac{1}{2}, P(A)>0, P(B)=b$ and $P(C)=c, P((\bar{B} \cap \bar{C} \mid A)=$

Two dice are thrown and the sum of the numbers appearing on the dice is observed to be a multiple of 4 . If $p$ is the conditional probability that number 4 has appeared atleast once, then $3 p+2=$

In a random experiment of throwing 5 coins, the number of heads is defined as a random variable. The mean of the random variable is

The variance of a Poisson variate $X$ is 2 . Then, $P(X \geq 3)=$

A cube having edge of length 5 cm is painted on all faces and then it is cut into equal cubes of unit volume. A small cube is selected at random and found that a face of it is painted, then the probability that two more faces of it are also painted is

A pair of dice is thrown twice in succession. The probability of getting prime number on both the dice in first throw and composite numbers on both the dice in second throw is

3 balls are drawn one after the other without replacement from an urn containing 4 red, 5 blue and 6 yellow balls. The probability of getting three different coloured balls is

Two balls are drawn at random from a bag containing 5 black balls and 3 white balls. If the random variable $X$ denotes the number of white balls drawn, then the mean of $X$ is

If the mean and variance of a binomial distribution are 4 and $\frac{4}{3}$ respectively, then $P(X=2)=$

4-digit numbers are formed using the digits 4, 5, 6, 7, 8, 9 allowing repetition of the given digits. If a number is chosen at random from those numbers thus formed, then the probability that it is exactly divisible by 3 is

If $E_1, E_2 \ldots, E_n$ are an independent events such that $P\left(E_r\right)=\frac{1}{1+r},(r=1,2, \ldots, n)$, then the probability that atleast one of $E_1, E_2, \ldots, E_n$ happens is

An urn contains five balls. Two balls are drawn at random and they are found to be white. The probability that all the balls in the urn are white, is

If the probability function of a random variable $X$ is given by $P(X=n)=\frac{k(n+1)}{3 n}$ for $n \in \mathbf{N} \cup\{0\}$ where $k$ is a constant, then $P(X<2)=$

An observer counts 240 vehicles per hour at a specific location on a highway. Assuming that the arrival of vehicles at the location follows Poisson distribution, the probability that more than two vehicles arrive over a 30 sec time interval is

If a man throws a die until he gets a number bigger than 3 , then the probability that he gets a 5 in his last throw is

A diagnostic test has the probability 0.95 of giving a positive result when applied to a person suffering from a certain disease and a probability 0.10 of giving a positive result when given to a non-sufferer. It is estimated that $0.5 \%$ of the population are suffering from the disease. If this test is now administered to a person from this population about whom there is no information relating to the incidence of this disease and the test gives a positive result, then the probability that he is a sufferer, is

Consider the following statements

Assertion (A) If $P_1, P_2, P_3$ are probability of happening of three independent events, then probability of happening of atleast one of them is $1-\left[\left(1-P_1\right)\left(1-P_2\right)\left(1-P_3\right)\right]$

Reason (R) For any three independent events $A, B$ and $C$

$ \begin{array}{r} P(A \cup B \cup C)=P(A)+P(B)+P(C)-P(A) P(B)-P(A) P(C) -P(B) P(C)+P(A) P(B) P(C) \end{array} $

The correct option among the following is

If probability function of a discrete random variable $X$ is $P(X=r)=r / k, r=1,2,3,4,5$, then $P\left(X=2\right.$ or $\left.X=\frac{k}{3}\right)$, is

If the probability that an individual will suffer a reaction from an injection of a drug is 0.001 , then the probability that out of 2000 individuals having that injection, more than 2 individuals will suffer a reaction, is

If $A_1, A_2, \ldots, A_{15}$ are the events of a random experiment, then which one of the following is true?

In an examination there are four Yes/No type of questions. The probability that the answer by the student to a question without guess to be correct is $2 / 3$. The probability that a student guesses a correct answer is $1 / 2$. A student writes the examination either by without guessing answers to all the 4 questions or by guessing answers to all 4 questions. The probability that he attempt the exam by guessing answers to all questions is $3 / 7$. Given that a student answered at least 3 questions correctly, the probability that he answered all the questions without guessing is

Four boxes $A, B, C$ and $D$ contain 5000, 3000, 2000 and 1000 fuses respectively. The percentages of defective fuses in these boxes are $3 \%, 2 \%, 1 \%$ and $0.5 \%$ respectively. If a fuse selected at random from one of the boxes is found to be defective, then the probability that it has come from box $D$ is

A die is thrown thrice. If getting 1 or 6 in a single throw is considered as success, then the variance of the number of successes is

In a hospital, on an average if there are 35 births in a weak, then the probability that there will be less than 3 births in a day, is

If $A$ and $B$ are events of a sample space such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}$ and $P(\bar{A})=\frac{2}{3}$, then $P(\bar{A} \cap B)$ is

Let $X$ and $Y$ be two events of a sample space such that $P(X)=\frac{1}{3}, P(X / Y)=\frac{1}{2}$ and $P(Y / X)=\frac{2}{5}$ then

Let $A$ and $B$ be not mutually exclusive events. If $P(A)=\frac{4}{9}, P(A \cap \bar{B})=\frac{3}{7}$ then $P\left(\frac{B}{A}\right)=$

If $20 \%$ of the bolts produced by a machine are defective then the probability that out of 4 bolts chosen at random, less than 2 bolts will be defective, is

In a book consisting of 600 pages, there are 60 typographical errors. The probability that a randomly chosen page will contain at most two errors, is