Probability

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 $\mu$ ...

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

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

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 $n^2 P(x>1)=$

$70 \%$ 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 t...

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 $P(\overline{\mathrm{~A}} \cap \mathrm{~B})=$

If the probability distribution of a random variable $X$ is as follows, then $k$ is equal to
$$ \begin{array}{c|l|l|l|l} \hline X=x & 1 & 2 & 3 & 4 \\ \hline P(X=x) & 2 k & 4 k & 3 k & k \\ \hline \end{array} $$

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

When two dice are thrown the probability of getting the sum of the values on them as 10 or 11 is

It is given that in a random experiment events $A$ and $B$ are such that $P(A)=\frac{1}{4}, P(A / B)=\frac{1}{2}$ and $P(B / A)=\frac{2}{3}$, then $P(B)$ is equal to

In a binomial distribution $B(n, p)$ the sum and product of the mean and the variance are 5 and 6 respectively, then $6(n+p-q)$ is equal to

Bag $A$ contains 2 white and 3 red balls and bag $B$ contains 4 white and 5 red balls. If one ball is drawn at random from one of the bags and is found to be red, then the probability that it was drawn from the bag $B$ is

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

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

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

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

Two persons $A$ and $B$ throw three unbiased dice one after the another. If $A$ gets the sum 13, then the probability that $B$ gets higher sum is

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

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

$E_1$ 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} \hlin...

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{arr...

$A$ and $B$ throw a pair of dice alternately and they note the sum of the numbers appearing on the dice. $A$ wins if he throws 6 before $B$ throws 7 and $B$ wins if he throws 7 before $A$ throws 6 . If $A$ begins then, the probability of hi...

If each of the coefficients $a, b$ and $c$ in the equation $a x^2+b x+c=0$ is determined by throwing a die, then the probability that the equation will have equal roots, is

A radar system can detect an enemy plane in one out of 10 consecutive scans. The probability that it cannot detect an enemy plane at least two times in four consecutive scans, is

7 coins are tossed simultaneously and the number of heads turned up is denoted by random variable $X$. If $\mu$ is the mean and $\sigma^2$ is the variance of $X$, then $\frac{\mu \sigma^2}{P(X=3)}=$

If 12 dice are thrown at a time, then the probability that a multiple of 3 does not appear on any dice is

A manufacturing company noticed that $1 \%$ of its products are defective. If a dealer order for 300 items from this company, then the probability that the number of defective items is atmost one is

A basket contains 12 apples in which 3 are rotten. If 3 apples are drawn at random simultaneously from it, then the probability of getting atmost one rotten apple is

In a class consisting of 40 boys and 30 girls. $30 \%$ of the boy and $40 \%$ of the girls are good at Mathematics. If a student selected at random from that class is found to be a girl, then the probability that she is not good at Mathemat...

If two numbers $x$ and $y$ are chosen one after the other at random with replacement from the set of number $\{1,2,3, \ldots \ldots 10\}$. Then, the probability that $\left|x^2-y^2\right|$ is divisible by 6 is

If $X \sim B(5, p)$ is a binomial variate such that $P(X=3)=P(X=4)$, then $P(|X-3|<2)=$

Bag $A$ contains 3 white and 4 red balls, bag $B$ contains 4 white and 5 red balls and bag $C$ is contains 5 white and 6 red balls. If one ball is drawn at random from each of these three bags, then the probability of getting one white and ...

If all the letters of the word 'SENSELESSNESS' are arranged in all possible ways and an arrangement among them is chosen at random, then the probability that all the E's come together in that arrangement is

An urn contains 3 black and 5 red balls. If 3 balls are drawn at random from the urn, the mean of the probability distribution of the number of red balls drawn is

Two persons $A$ and $B$ throw a pair of dice alternately until one of them gets the sum of the numbers appeared on the dice as 4 and the person who gets this result first is declared as the winner. If $A$ starts the game, then the probabili...

$P(A \mid A \cap B)+P(B \mid A \cap B)=$

In a random experiment, two dice are thrown and the sum of the numbers appeared on them is recorded. This experiment is repeated 9 times. If the probability that a sum of 6 appears atleast once is $P_1$ and a sum of 8 appears atleast once i...

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

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For the probability distribution of a discrete random variable $X$ as given below, then mean of $X$ is

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Two digits are selected at random from the digits 1 through 9. If their sum is even, then the probability that both are odd, is

A, B and C are mutually exclusive and exhaustive events of a random experiment and $E$ is an event that occurs in conjunction with one of the events $\mathrm{A}, \mathrm{B}$ and $C$. The conditional probabilities of $E$ given the happening ...

Two natural numbers are chosen at random from 1 to 100 and are multiplied. If $A$ is the event that the product is an even number and $B$ is the event that the product is divisible by 4 , then $P(A \cap \bar{B})=$

If five-digit numbers are formed from the digits $0,1,2,3,4$ using every digit exactly only once. Then, the probability that a randomly chosen number from those numbers is divisible by 4 is

The probability that a man failing to hit a target is $\frac{1}{3}$. If he fires 4 times, then the probability that he hits the target at least thrice is

A person is known to speak false once out of 4 times, If that person picks a card at random from a pack of 52 cards and reports that it is a king, then the probability that it is actually a king is

A box $P$ contains one white ball, three red ball and two black balls. Another box $Q$ contains two white balls, three red balls and four black balls. If one ball is drawn at random from each one of the two boxes, then the probability that ...

For a binomial variate $X \sim B(n, p)$ the difference between the mean and variance is 1 and the difference between their square is 11 . If the probability of $P(x=2)=m\left(\frac{5}{6}\right)^n$ and $n=36$, then $m: n$

If two cards are drawn randomly from a pack of 52 playing cards, then the mean of the probability distribution of number of kings is

If 7 different balls are distributed among 4 different boxes, then the probability that the first box contains 3 balls is