Three numbers are chosen from 1 to 30 . The probability that they are not three consecutive numbers is

If two events $A$ and $B$ are such that $P(\bar{A})=03, P(B)=0.4$ and $P(A \cap \bar{B})=0.5$, then $P(B / A \cup \bar{B})=$

Two candidates $A$ and $B$ have attended an interview conducted by a recruitment board for two jobs, If the probability that candidate $A$ will get the job is 0.8 and the probability that candidate $B$ will get the job is 0.7 , then the probability that atleast one of them will get the job is

X denotes the number of times heads that occur in $n$ tosses of a fair coin. If $P(X=4), P(X=5)$ and $P(X=6)$ ate in arithmetic progression. The largest value of $n$ is

The probability distribution of a random variable $X$ is as follows. Then, the mean of $x$ is

Two students appeared simultaneously for an entrance exam. If the probability that the first student gets qualified in the exam is $\frac{1}{4}$ and the probability that the second student gets qualified in the same exam is $\frac{2}{5}$, then the probability that atleast one of them gets qualified in that exam is

For three events $A, B$ and $C$ of a sample space, $P$ (exactly one of $A$ or $B$ occurs ) $=P$ (exactly one of $B$ or $C$ occurs) $=P($ exactly one of $C$ or $A$ occurs $)=\frac{1}{4}$. If probability of all the three events occurring simultaneously is $\frac{1}{16}$, then the probability that atleast one of the events occur is

$A$ bag $P$ contains 4 red and 5 black balls another bag Q contains 3 red and 6 black balls. If one ball is drawn at random from bag $P$ and two balls are drawn from bag $Q$, then the probability that out of the three balls drawn two are black and one is red, is

On every evening, a student either watches TV or reads a book. The probability of watching TV is $\frac{4}{5}$ If he watches TV, the probability that he will fall asleep is $\frac{3}{4}$ and it is $\frac{1}{4}$ when he reads a book. If the student is found to be asleep on an evening the probability that he watched the TV is

Let $X$ be the random variable taking values $1,2, \ldots n$ for a fixed positive integer $n$. If $P(X=k)=\frac{1}{n}$ for $1 \leq k \leq n$, then the variance of $X$ is

A radar system can detect an enemy plane in one out of ten consecutive scans.

The probability that it can detect an enemy plane atleast twice in four consecutive scans is

A company representative is distributing 5 identical samples of a product among 12 houses in a row such that each house gets at most one sample. The probability that no two consecutive house get one sample is

38. $A$ and $B$ are two independent events of a random experiment and $P(A)>P(B)$.

If the probability that both $A$ and $B$ occurs is $\frac{1}{6}$ and neither of them occurs is $\frac{1}{3}$, then the probability of the occurance of $B$ is

Two dice are thrown and the sum of the numbers appeared on the dice is noted. If $A$ is the event of getting a prime number as their sum and $B$ is the event of getting a number greater than 8 as their sum, then $P(A \cap \bar{B})=$

A family consists of 8 persons. If 4 persons are chosen a random and they are found to be 2 men and 2 women, then the probability that there are equal number of men and women in that family is

The number of trials conducted in a binomial distribution is 6 . If the difference between the mean and variance of this variate is $\frac{27}{8}$, then the probability of getting atmost 2 successes is

Let $X \sim B(n, p)$ with mean $\mu$ and variance $\sigma^2$. If $\mu=2 \sigma^2$ and $\mu+\sigma^2=3$, then $P(X \leq 3)=$

A basket contains 5 apples and 7 oranges and another basket contains 4 apples and 8 oranges. If one fruit is picked out at random from each basket, then the probability of getting one apple and one orange is

Two cards are drawn from a pack of 52 playing cards one after the other without replacement. If the first card drawn is a queen, then the probability of getting a face card from a black suit in the second draw is

An item is tested on a device for its defectiveness. The probability that such an item is defective is 0.3 . The device gives accurate result in 8 out of 10 such tests.

If the device reports that an item tested is not defective, then the probability that it is actually defective is

In a school there are 3 sections $A, B$ and $C$. Section $A$ contains 20 girls and 30 boys, section $B$ contains 40 girls and 20 boys and section $C$ contains 10 girls and 30 boys. The probabilities of selecting the section $A, B$ and $C$ are $0.2,0.3$ and 0.5 respectively. If a student selected at random from the school is a girl, then the probability that she belongs to section $A$ is

If the probability distribution of a random variable $X$ is as follows, then the mean of $X$ is

$ \begin{array}{ccccc} \hline \boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}} & -1 & 0 & 1 & 2 \\ \hline \boldsymbol{P}\left(\boldsymbol{X}=\boldsymbol{x}_{\boldsymbol{i}}\right) & \boldsymbol{k}^3 & 2 \boldsymbol{k}^3+\boldsymbol{k} & 4 \boldsymbol{k}-10 \boldsymbol{k}^2 & 4 \boldsymbol{k}-1 \\ \hline \end{array} $

If $X$ is a binomial variate with mean $\frac{16}{5}$ and variance $\frac{48}{25}$, then $P(X \leq 2)=$

A die is thrown twice. Let A be the event of getting a prime number when the die is thrown first time and $B$ be the event of getting an even number when the die is thrown second time. Then, $P(A / \bar{B})=$

A bag contains 5 balls of unknown colours. There are equal chances that out of these five balls, there may be 0 or 12 or or 3 or 4 or 5 red balls, A ball is taken out from the bag at random and is found to be red. The probability that it is the only red ball in the bag is

If $X \sim B(9, p)$ is a binomial variate satisfying the equation $P(X=3)=P(X=6)$, then $P(X<3)=$

The probability distribution of a discrete random variable $X$ is given below

$ \begin{array}{lllll} \hline X=x & -1 & 0 & 1 & 2 \\ \hline P(X=x) & \frac{1}{3} & \frac{1}{6} & \frac{1}{6} & \frac{1}{3} \\ \hline \end{array} $

Then, the value of $6 \sum\left(x^2\right) P(X=x)-\operatorname{var}(X)=$

If the average number of accidents occurring at a particular junction on a highway in a week is 5 , then the probability that atmost one accident occurs in a particular week is

An unbiased coin is tossed 8 times. The probability that head appears consecutively at least 5 times is

A box contains twelve balls of which 4 are red, 5 are green and 3 are white. If three balls are drawn at random simultaneously from the box, then the probability that exactly 2 balls have the same colour is

There are three families $F_1, F_2, F_3 . F_1$ has 2 boys and 1 girl; $F_2$ has 1 boy and 2 girls; $F_3$ has 1 boy and 1 girl. A family is randomly chosen and a child is chosen from that family randomly. If it is known that the child thus selected is a girl, then the probability that she is form $F_2$ is

An urn $A$ contains 4 white and 1 black ball; urn $B$ contains 3 white and 2 black balls and urn $C$ contains 2 white and 3 black balls. One ball is transferred randomly from $A$ to $B$; later one ball is transferred randomly from $B$ to $C$. Finally, if a ball is drawn randomly from $C$, then the probability that it is a black ball is

In a binomial distribution, if $n=4$ and $P(X=0)=\frac{16}{81}$, then $P(X=4)=$

If $l, m$ represent any two elements (identical or different) of the set $\{1,2,3,4,5,6,7\}$, then the probability that $l x^2+m x+1>0 \forall x \in R$ is

$A$ and $B$ are playing chess game with each other. The probability that $A$ wins the game is 0.6 . the probability that he loses is 0.3 and the probability its draw is 0.1 . If they played three games, then the probability that $A$ wins atleast two games is

$U_1, U_2, U_3$ are three urns. $U_1$ contains 5 red, 3 white, 2 back balls: $U_2$ contains 4 red 4 white, 2 black balls and $U_3$ contains 3 red. 4 white, 3 black balls. If a ball is chosen at random from an urn chosen at random, then the probability of not getting a black ball is

If the probability distribution of a random variable $X$ is as follows, then $P(X \leq 2)=$

$ \begin{array}{cccccc}\hline x_i & 0 & 1 & 2 & 3 & 4 \\ \hline P\left(X=x_i\right) & 3 k & 5 k & 3 k^2 & 4 k^2+k & 3 k^2 \\ \hline \end{array} $

If $X$ follows poisson distribution with variance 2 , then $P(X \geq 3)=$

A problem in Algebra is given to two students $A$ and $B$ whose chances of solving it are $\frac{2}{5}$ and $\frac{3}{4}$ respectively.

The probability that the problem is solved if both of them try independently is

Three dice are thrown simultaneously and the sum of the numbers appeared on them is noted. If $A$ is the event of getting a sum greater than 14 and $B$ is the event of getting a sum which is a multiple of 3 , then $P(A \cap \bar{B})+P(\bar{A} \cap B)=$

A manufacturing company of bulbs has 3 units $A, B$ and $C$ which produce $25 \%, 35 \%$ and $40 \%$ of the bulbs respectively. Out of the bulbs produced by $A, B, C$ units, $5 \%, 4 \%$ and $2 \%$ are defective, respectively. If a bulb is chosen at random and found to be defective, then the probability that it is produced by unit $B$ is

The probability distribution of a random variable $X$ is given below

$ \begin{array}{ccccccc} \hline X & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline P\left(X=x_i\right) & \alpha & \alpha & \alpha & \beta & \beta & 0.3 \\ \hline \end{array} $

If $\mu$ and $\sigma^2$ represent the mean and variance of $X$ and $\mu=4.2$, then $\sigma^2+\mu^2=$

The probability that a student gets distinction in a Mathematics test is $\frac{2}{3}$. If five such tests are conducted over a certain period of time, then the probability that he gets distinction in atleast 3 tests is

If $A$ and $B$ are events of a random experiment such that $P(A \cup B)=\frac{3}{4}, P(A \cap B)=\frac{1}{4}, P(\overline{\mathrm{~A}})=\frac{2}{3}$, then $P(\overline{\mathrm{~A}} \cap \mathrm{~B})=$

Two cards are drawn at random from a pack of 52 playing cards. If both the cards drawn are found to be black in colour, then the probability that atleast one of them is face card is

A person is known to speak the truth in 3 out of 4 occasions. If he throws a die and reports that it is six, then the probability that it actually six is

$70 \%$ of the total employees of a factory are men. Among the employees of that factory 30\% of men and $15 \%$ of women are technical assistants. If an employee chosen at random is found to be a technical assistant, then the probability that this employee is a man is

If a discrete random variable $X$ has the probability distribution $P(X=x)=k \frac{2^{2 x+1}}{(2 x+1)!}, x=0,1,2 \ldots \infty$, then $k=$

A random variable $X$ follows a binomial distribution in which the difference between its mean and variance is 1. if $2 P(x=2)=3 P(x=1)$, then $n^2 P(x>1)=$

The probability that $A$ speaks truth is $75 \%$ and the probability that $B$ speaks truth is $80 \%$. The probability that they contradict each other when asked to speak on a fact is

If the probability distribution of a random variable $X$ is as follows, then $k$ is equal to

$ \begin{array}{c|l|l|l|l} \hline X=x & 1 & 2 & 3 & 4 \\ \hline P(X=x) & 2 k & 4 k & 3 k & k \\ \hline \end{array} $

$E_1$ and $E_2$ are two independent events of a random experiment such that $P\left(E_1\right)=\frac{1}{2}$ and $P\left(E_1 \cup E_2\right)=\frac{2}{3}$. Then, match the items of List I with the items of List II.

$ \begin{array}{lll} \hline & \text { List I } & \text { List II } \\ \hline \text { (A) } & P\left(E_2\right) & \text { (i) }1/2 \\ \hline \text { (B) } & P\left(E_1 / E_2\right) & \text { (ii) } 5 / 6 \\ \hline \text { (C) } & P\left(E_2 / E_1\right) & \text { (iii) } 1 / 3 \\ \hline \text { (D) } & P\left(E_1 \cup E_2\right) & \text { (iv) } 1 / 6 \\ \hline & & \text { (v) } 2 / 3 \\ \hline \end{array} $

A bag contains 4 red and 5 black balls. Another bag contains 3 red and 6 black balls. If one ball is drawn from first bag and two balls from the second bag at random. The probability that out of the three, two are black and one is red, is

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

$ \begin{array}{clllllll} \hline X=x & -3 & -2 & -1 & 0 & 1 & 2 & 3 \\ \hline P(X=x) & 0.05 & 0.1 & 2 K & 0 & 0.3 & K & 0.1 \\ \hline \end{array} $

Three numbers are chosen at random from 1 to 20 , then the probability that the sum of three numbers is divisible by 3 is

8 teachers and 4 students are sitting around a circular table at random, then the probability that no two students sit together is

A bag contains 6 balls. If three balls are drawn at a time and all of them are found to be green, then the probability that exactly 5 of the balls in the bag are green is

In a binomial distribution the difference between the mean and standard deviation is 3 and the difference between their squares is 21 , then $P(x=1): P(x=2)=$

When an unfair dice is thrown the probability of getting a number $k$ on it is $P(X=k)=k^2 P$, where $k=1,2,3,4,5,6$ and $X$ is the random variable denoting a number on the dice, then the mean of X is

S is the sample space and $A, B$ are two events of a random experiment. Match the items of List $A$ with the items of List B

The probability of getting a sum 9 when two dice are thrown is

If $A$ and $B$ are two events such that $P(B) \neq 0$ and $P(B) \neq 1$, then $P(\bar{A} \mid \bar{B})$ is

Two brothers $X$ and $Y$ appeared for an exam. Let $A$ be the event that $X$ has passed the exam and $B$ is the event that $Y$ has passed. The probability of $A$ is $\frac{1}{7}$ and of $B$ is $\frac{2}{9}$. Then, the probability that both of them pass the exam is

A bag contains 4 red and 3 black balls. A second bag contains 2 red and 3 black balls. One bag is selected at random. If from the selected bag, one ball is drawn at random, then the probability that the ball drawn is red, is

In a Binomial distribution, if '$n$' is the number of trials and the mean and variance are 4 and 3 respectively, then $2^{32} p\left(X=\frac{n}{2}\right)=$

For a Poisson distribution, if mean $=l$, variance $=m$ and $l+m=8$, then $e^4[1-P(X>2)]=$

In a box, there are 8 red, 7 blue and 6 green balls. One ball is picked randomly. The probability that it is neither red nor green is

For two events $A$ and $B$, a true statement among the following is

Five digit numbers are formed by using digits $1,2,3,4$ and 5 without repetition. Then, the probability that the randomly chosen number is divisible by 4 is

A manager decides to distribute ₹ 20000 between two employees $X$ and $Y$. He knows $X$ deserves more than $Y$, but does not know how much more. So, he decides to arbitrarily break ₹ 20000 into two parts and give $X$ the bigger part. Then, the chance that $X$ gets twice as much as $Y$ or more is

Which of the following is not a property of a Binomial distribution?

In a Binomial distribution $B(n, p)$, if the mean and variance are 15 and 10 respectively, then the value of the parameter $n$ is

A box contains 100 balls, numbered from 1 to 100 . If 3 balls are selected one after the other at random with replacement from the box, then the probability that the sum of the three numbers on the balls selected from the box is an odd number, is

In a lottery, containing 35 tickets, exactly 10 tickets bear a prize. If a ticket is drawn at random, then the probability of not getting a prize is

A bag contains 7 green and 5 black balls. 3 balls are drawn at random one after the other. If the balls are not replaced, then the probability of all three balls being green is

If $x$ is chosen at random from the set $\{1,2,3, 4\}$ and $y$ is chosen at random from the set $\{5,6,7\}$, then the probability that $x y$ will be even is

The discrete random variables $X$ and $Y$ are independent from one another and are defined as $X \sim B(16,0.25)$ and $Y \sim P(2)$. Then, the sum of the variance of $X$ and $Y$ is

If 6 is the mean of a Poisson distribution, then $P(X \geq 3)=$

A coin is tossed until a head appears or it has been tossed thrice. Given that head doesn’t appear on the first toss, the probability that coin tossed thrice is

Box-I contains 3 cards bearing numbers 1, 2, 3 , Box II contains 5 cards bearing numbers 1 , 2, 3, 4, 5 and Box III contains 7 cards bearing numbers 1, 2, 3, 4, 5, 6, 7. One card is drawn at random from each of the boxes. If $x_i$ be the number on the card drawn from the $i$ th box, $i=1,2,3$, then the probability that $x_1+x_2+x_3$ is odd is equal to

The range of a random variable $X$ is $\{1,2,3, \ldots\}$ and $P(X=x)=\frac{C^x}{x !}$. for $x=1,2,3$, ... Then, the value of $C$ is

Tom and Jerry play a game of alternately throwing an unfair coin. First one to get head wins. If Tom starts the game, he has 62.5% chance of winning the game. Suppose this coin is tossed 5 times, then the probability of getting exactly 3 head is

One card is selected at random from 27 cards numbered form 1 to 27. What is the probability that the number on the card is even or divisible by 5.

Nine balls one drawn simultaneously from a bag containing 5 white and 7 black balls. The probability of drawing 3 white and 6 black balls is

The probabilities that $A$ and $B$ speak truth are $\frac{4}{5}$ and $\frac{3}{4}$ respectively. The probability that they contradict each other when asked to speak on a fact is

The mean and variance of a binomial variable X are 2 and 1 respectively. The probability that X takes values greater than 1 is

P speaks truth in 70% of the cases and Q in 80% of the cases. In what percent of cases are they likely to agree in stating the same fact

If $A$ and $B$ are two events with $P(A \cap B)=\frac{1}{3}, P(A \cup B)=\frac{5}{6}$ and $P\left(A^C\right)=\frac{1}{2}$, then the value of $P\left(B^C\right)$ is

A coin is tossed 2020 times. The probability of getting head on 1947th toss is

A discrete random variable X takes values 10, 20, 30 and 40. with probability 0.3, 0.3, 0.2 and 0.2 respectively. Then the expected value of X is

Let $X$ be a random variable which takes values $1,2,3,4$ such that $P(X=r)=K r^3$ where $r=1,2,3,4$ then

12 balls are distributed among 3 boxes, then the probability that the first box will contain 3 balls is

A random variable X has the probability distribution

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

A die is tossed thrice. If event of getting an even number is a success, then the probability of getting at least 2 successes is