Probability · Mathematics · MHT CET

If a discrete random variable X is defined as follows $\mathrm{P}[\mathrm{X}=x]=\left\{\begin{array}{cl}\frac{\mathrm{k}(x+1)}{5^x}, & \text { if } x=...

Numbers are selected at random, one at a time from two digit numbers $10,11,12 \ldots ., 99$ with replacement. An event $E$ occurs if and only if the ...

Two friends A and B apply for a job in the same company. The probabilities of A getting selected is $\frac{2}{5}$ and that of B is $\frac{4}{7}$. Then...

If a random variable X has the following probability distribution values .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;...

A random variable X takes the values $0,1,2,3$ and its mean is 1.3 . If $\mathrm{P}(\mathrm{X}=3)=2 \mathrm{P}(\mathrm{X}=1)$ and $P(X=2)=0.3$, then $...

Three persons $\mathrm{P}, \mathrm{Q}$ and R independently try to hit a target. If the probabilities of their hitting the target are $\frac{3}{4}, \fr...

A box contains 15 green and 10 yellow balls. If 10 balls are randomly drawn, one by one, with replacement, then the variance of the number of green ba...

 Suppose three coins are tossed simultaneously. If $X$ denotes the number of heads, then probability distribution of x is

Four fair dice are thrown independently 27 times. Then the expected number of times, at least two dice show up a three or a five is

If two fair dice are rolled, then the probability that the sum of the numbers on the upper faces is at least 9, is

For the probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font...

A random variable X has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-sty...

The probability, that a year selected at random will have 53 Mondays, is

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a stu...

A random variable $X$ has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-s...

Let $\mathrm{A}, \mathrm{B}$ and C be three events, which are pairwise independent and $\bar{E}$ denote the complement of an event E . If $\mathrm{P}(...

A random variable x takes the values $0,1,2$, $3, \ldots$ with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1)\left(\frac{1}{5}\right)^x$, where...

One hundred identical coins, each with probability p , of showing up heads are tossed once. If $0...

The p.m.f. of a random variable X is given by $$\begin{aligned} \mathrm{P}[\mathrm{X}=x] & =\frac{\binom{5}{x}}{2^5}, \text { if } x=0,1,2,3,4,5 \\ & ...

If three fair coins are tossed, then variance of number of heads obtained, is

If $A$ and $B$ are two independent events such that $\mathrm{P}\left(\mathrm{A}^{\prime}\right)=0.75, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=0.65$ and...

The probability that a person who undergoes a bypass surgery will recover is 0.6 . the probability that of the six patients who undergo similar operat...

$A$ and $B$ are independent events with $P(A)=\frac{3}{10}$, $\mathrm{P}(\mathrm{B})=\frac{2}{5}$, then $\mathrm{P}\left(\mathrm{A}^{\prime} \cup \mat...

Minimum number of times a fair coin must be tossed, so that the probability of getting at least one head, is more than $99 \%$ is

A random variable X assumes values $1,2,3, \ldots \ldots ., \mathrm{n}$ with equal probabilities. If $\operatorname{var}(X): E(X)=4: 1$, then $n$ is e...

In a game, 3 coins are tossed. A person is paid ₹ 100$, if he gets all heads or all tails; and he is supposed to pay ₹ 40 , if he gets one head or two...

In a Binomial distribution consisting of 5 independent trials, probabilities of exactly 1 and 2 successes are 0.4096 and 0.2048 respectively, then the...

A random variable X has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-sty...

 There are three events $\mathrm{A}, \mathrm{B}, \mathrm{C}$, one of which must and only one can happen. The odds are 8:3 against $\mathrm{A}, 5: 2$ a...

Four persons can hit a target correctly with probabilities $\frac{1}{2}, \frac{1}{3}, \frac{1}{4}$ and $\frac{1}{5}$ respectively. If all hit at the t...

If the mean and the variance of a Binomial variate X are 2 and 1 respectively, then the probability that X takes a value greater than one is equal to...

A bag contains 4 red and 3 black balls. One ball is drawn and then replaced in the bag and the process is repeated. Let X denote the number of times b...

A service station manager sells gas at an average of ₹ 100 per hour on a rainy day, ₹ 150 per hour on a dubious day, ₹ 250 per hour on a fair day and ...

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. ...

A random variable X takes values $-1,0,1,2$ with probabilities $\frac{1+3 \mathrm{p}}{4}, \frac{1-\mathrm{p}}{4}, \frac{1+2 \mathrm{p}}{4}, \frac{1-4 ...

If the mean and the variance of Binomial variate $X$ are 2 and 1 respectively, then the probability that X takes a value greater than or equal to one ...

If $\mathrm{P}(\mathrm{X}=2)=0.3, \mathrm{P}(\mathrm{X}=3)=0.4, \mathrm{P}(\mathrm{X}=4)=0.3$, then the variance of random variable X is

A man and his wife appear for an interview for two posts. The probability of the husband's selection is $\frac{1}{7}$ and that of the wife's selection...

The expected value of the sum of the two numbers obtained on the uppermost faces, when two fair dice are rolled, is

For an entry to a certain course, a candidate is given twenty problems to solve. If the probability that the candidate can solve any problem is $\frac...

A random variable has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style...

Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{...

A random variable $X$ has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-s...

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability, that a st...

Let a random variable X have a Binomial distribution with mean 8 and variance 4 . If $\mathrm{P}(x \leqslant 2)=\frac{\mathrm{k}}{2^{16}}$, then k is ...

For the probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-width:1px;font...

Two cards are drawn successively with replacement from a well- shuffled pack of 52 cards. Let X denote the random variable of number of kings obtained...

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

A random variable x has the following probability distribution. Then value of $k$ is _________ and $\mathrm{P}(3 .tg {border-collapse:collapse;borde...

Let $\mathrm{X} \sim \mathrm{B}\left(6, \frac{1}{2}\right)$, then $\mathrm{P}[|x-4| \leqslant 2]$ is

A person throws an unbiased die. If the number shown is even, he gains an amount equal to the number shown. If the number is odd, he loses an amount e...

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Let X denote the random variable of number of jacks obtained ...

Let in a Binomial distribution, consisting of 5 independent trials, probabilities of exactly 1 and 2 successes be 0.4096 and 0.2048 respectively. Then...

The probability distribution of a random variable X is given by .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-st...

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability th...

A bag contains 4 Red and 6 Black balls. A ball is drawn at random from the bag, its colour is observed and this ball along with 3 additional balls of ...

If a discrete random variable X takes values $0,1,2,3, \ldots \ldots$. with probability $\mathrm{P}(\mathrm{X}=x)=\mathrm{k}(x+1) 5^{-x}$, where k is ...

Ten bulbs are drawn successively, with replacement, from a lot containing $10 \%$ defective bulbs, then the probability that there is at least one def...

A fair die with numbers 1 to 6 on their faces is thrown. Let $$\mathrm{X}$$ denote the number of factors of the number, on the uppermost face, then th...

The p.m.f. of a random variable $$\mathrm{X}$$ is $$\mathrm{P}(x)=\left\{\begin{array}{cl}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \mat...

There are 6 positive and 8 negative numbers. From these four numbers are chosen at random and multiplied. Then the probability, that the product is a ...

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs are selected at random from the lot and are sent to retail store. Then the probability that...

Two cards are drawn successively with replacement from well shuffled pack of 52 cards, then the probability distribution of number of queens is

For an initial screening of an entrance exam, a candidate is given fifty problems to solve. If the probability that the candidate can solve any proble...

$$\text { If } f(x)= \begin{cases}3\left(1-2 x^2\right) & ; 0...

Three critics review a book. For the three critics the odds in favour of the book are $$2: 5, 3: 4$$ and $$4: 3$$ respectively. The probability that t...

Two dice are rolled. If both dice have six faces numbered $$1,2,3,5,7,11$$, then the probability that the sum of the numbers on upper most face is pri...

A random variable $$X$$ has the probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:sol...

A random variable $$X$$ has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border...

Let $$\mathrm{X}$$ be random variable having Binomial distribution $$B(7, p)$$. If $$P[X=3]=5 P[X=4]$$, then variance of $$\mathrm{X}$$ is

If a continuous random variable $$\mathrm{X}$$ has probability density function $$\mathrm{f}(x)$$ given by $$f(x)=\left\{\begin{array}{cl} a x & , \te...

A card is drawn at random from a well shuffled pack of 52 cards. The probability that it is black card or face card is

An irregular six faced die is thrown and the probability that, in 5 throws it will give 3 even numbers is twice the probability that it will give 2 ev...

A box contains 100 tickets numbered 1 to 100 . A ticket is drawn at random from the box. Then the probability, that number on the ticket is a perfect ...

Three fair coins with faces numbered 1 and 0 are tossed simultaneously. Then variance (X) of the probability distribution of random variable $$\mathrm...

The p.m.f of random variate $$\mathrm{X}$$ is $$P(X)= \begin{cases}\frac{2 x}{\mathrm{n}(\mathrm{n}+1)}, & x=1,2,3, \ldots \ldots, \mathrm{n} \\ 0, & ...

An experiment succeeds twice as often as it fails. Then the probability, that in the next 6 trials there will be atleast 4 successes, is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then the probability distribution of number of jacks is...

$$\mathrm{A}$$ and $$\mathrm{B}$$ are independent events with $$\mathrm{P}(\mathrm{A})=\frac{1}{4}$$ and $$\mathrm{P}(\mathrm{A} \cup \mathrm{B})=2 \m...

Two cards are drawn successively with replacement from a well-shuffled pack of 52 cards. Then mean of number of tens is

A fair die is tossed twice in succession. If $$\mathrm{X}$$ denotes the number of fours in two tosses, then the probability distribution of $$\mathrm{...

If $$\mathrm{A}$$ and $$\mathrm{B}$$ are two events such that $$\mathrm{P}(\mathrm{A})=\frac{1}{3}, \mathrm{P}(\mathrm{B})=\frac{1}{5}, \mathrm{P}(\ma...

Let a random variable $$\mathrm{X}$$ have a Binomial distribution with mean 8 and variance 4. If $$\mathrm{P}(\mathrm{X} \leq 2)=\frac{\mathrm{K}}{2^{...

From a lot of 20 baskets, which includes 6 defective baskets, a sample of 2 baskets is drawn at random one by one without replacement. The expected va...

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with these three vertices is equilateral, equals __...

A binomial random variable $$\mathrm{X}$$ satisfies $$9. p(X=4)=p(X=2)$$ when $$n=6$$. Then $$p$$ is equal to

The three ships namely A, B and C sail from India to Africa. If the odds in favour of the ships reaching safely are $$2: 5,3: 7$$ and $$6: 11$$ respec...

If the sum of mean and variance of a Binomial Distribution is $$\frac{15}{2}$$ for 10 trials, then the variance is

In a game, 3 coins are tossed. A person is paid ₹ 7 /-, if he gets all heads or all tails; and he is supposed to pay ₹ 3 /-, if he gets one head or tw...

A fair die is tossed twice in succession. If $$\mathrm{X}$$ denotes the number of sixes in two tosses, then the probability distribution of $$\mathrm{...

For a binomial variate $$\mathrm{X}$$ with $$\mathrm{n}=6$$ if $$P(X=4)=\frac{135}{2^{12}}$$, then its variance is

The p.d.f. of a discrete random variable is defined as $$\mathrm{f}(x)=\left\{\begin{array}{l} \mathrm{k} x^2, 0 \leq x \leq 6 \\ 0, \text { otherwise...

A player tosses 2 fair coins. He wins ₹5 if 2 heads appear, ₹ 2 if one head appears and ₹ 1 if no head appears. Then the variance of his winning amoun...

Three critics review a book. For the three critics the odds in favor of the book are $$2: 5, 3: 4$$ and $$4: 3$$ respectively. The probability that th...

A man takes a step forward with probability 0.4 and backwards with probability 0.6 . The probability that at the end of eleven steps, he is one step a...

A problem in statistics is given to three students A, B and C. Their probabilities of solving the problem are $$\frac{1}{2}, \frac{1}{3}$$ and $$\frac...

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards, then mean of number of queens is

In a Binomial distribution with $$\mathrm{n}=4$$, if $$2 \mathrm{P}(\mathrm{X}=3)=3 \mathrm{P}(\mathrm{X}=2)$$, then the variance is

$$\mathrm{A}, \mathrm{B}, \mathrm{C}$$ are three events, one of which must and only one can happen. The odds in favor of $$\mathrm{A}$$ are $$4: 6$$, ...

The probability mass function of random variable X is given by $$P[X=r]=\left\{\begin{array}{ll} \frac{{ }^n C_r}{32}, & n, r \in \mathbb{N} \\ 0, & ...

Three fair coins numbered 1 and 0 are tossed simultaneously. Then variance Var (X) of the probability distribution of random variable $$\mathrm{X}$$, ...

The incidence of occupational disease in an industry is such that the workmen have a $$10 \%$$ chance of suffering from it. The probability that out o...

If $$P(A \cup B)=0.7, P(A \cap B)=0.2$$, then $$P\left(A^{\prime}\right)+P\left(B^{\prime}\right)$$ is

Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Then mean of number of kings is

The probability that at least one of the events $$E_1$$ and $$E_2$$ occurs is 0.6. If the simultaneous occurrence of $$\mathrm{E}_1$$ and $$\mathrm{E}...

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

The probability distribution of a random variable X is .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid...

A fair coin is tossed for a fixed number of times. If probability of getting 7 heads is equal to probability of getting 9 heads, then probability of g...

If the probability distribution function of a random variable X is given as .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:bla...

If $$\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{2}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{5}$$, then $$\mathrm{P}(...

The probability distribution of a discrete random variable X is .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-st...

A die is thrown four times. The probability of getting perfect square in at least one throw is

A man is known to speck truth 3 out of 4 times. He throws a die and reports that it is 6. Then the probability that it is actually 6 is

The mean of the numbers obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is

Let two cards are drawn at random from a pack of 52 playing cards. Let X be the number of aces obtained. Then the value of E(X) is

A fair coin is tossed 100 times. The probability of getting a head for even number of times is

If the function defined by $$f(x)=K(x-x^2)$$ if $$0 ...

The probability distribution of the number of doublets in four throws of a pair of dice is given by

For the probability distribution given by following .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;bo...

A random variable X has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-sty...

Rooms in a hotel are numbered from 1 to 19. Rooms are allocated at random as guests arrive. The first guest to arrive is given a room which is a prime...

The distribution function $$F(X)$$ of discrete random variable $$X$$ is given by .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-colo...

First bag contains 3 red and 5 black balls and second bag contains 6 red and 4 black balls. A ball is drawn from each bag. The probability that one ba...

A fair coin is tossed 4 times. If $$X$$ is a random variable which indicates number of heads, then $$\mathrm{P}[\mathrm{X}

If the mean and variance of a binomial distribution are 4 and 2 respectively, then probability of getting 2 heads is

For two events $$\mathrm{A}$$ and $$\mathrm{B}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{5}{6}, \mathrm{P}(\mathrm{A})=\frac{1}{6}, \mathrm{P}(\ma...

A random variable X has following distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:solid;border-...

A coin is tossed three times. If X denotes the absolute difference between the number of heads and the number of tails, then P (X = 1) =

A random variable X $$\sim$$ B (n, p), if values of mean and variance of X are 18 and 12 respectively, then n =

an urn contains 9 balls of which 3 are red, 4 are blue and 2 are green. Three balls are drawn at random from the urn. The probability that the three b...

It is observed that $$25 \%$$ of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the pr...

In a meeting $$60 \%$$ of the members favour and $$40 \%$$ oppose a certain proposal. A member is selected at random and we take $$\mathrm{X}=0$$ if h...

A lot of 100 bulbs contains 10 defective bulbs. Five bulbs selected at random from the lot and sent to retain store, then the probability that the sto...

A coin is tossed and a die is thrown. The probability that the outcome will be head or a number greater than 4 or both, is

If $$\mathrm{X}$$ is a random variable with p.m.f. as follows. $$\begin{aligned} \mathrm{P}(\mathrm{X}=\mathrm{x}) & =\frac{5}{16}, \mathrm{x}=0,1 \\ ...

The p.m.f. of a random variable X is $$\mathrm{P(X = x) = {1 \over {{2^5}}}\left( {_x^5} \right),x = 0,1,2,3,4,5}=0$$ then

The variance of the following probability distribution is, ...

If the sum of mean and variance of a binomial distribution for 5 trials is 1.8, then probability of a success is

Two unbiased dice are thrown. Then the probability that neither a doublet nor a total of 10 will appear is

Two dice are rolled simultaneously. The probability that the sum of the two numbers on the dice is a prime number, is

A random variable X has the following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-sty...

Rajesh has just bought a VCR from Maharashtra Electronics and the shop offers after sales service contract for Rs. 1000 for the next five years. Consi...

If $$X \sim B(4, p)$$ and $$P(X=0)=\frac{16}{81}$$, then $$P(X=4)=$$

The odds in favour of getting sum multiple of 3 , when pair of dice are thrown is

If $X$ is a.r.v. with c.d.f $F(x)$ and its probability distribution is given by .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color...

The odds in favour of drawing a king from a pack of 52 playing cards is

Out of 100 people selected at random, 10 have common cold. If five persons selected at random from the group, then the probability that at most one pe...

The pdf of a continuous r.v. $$X$$ is given by $$f(x)=\frac{x}{8}, 0

If a die is thrown at random, then the expectation of the number on it is

The probability that bomb will miss the target is 0.2. Then, the probability that out of 10 bombs dropped exactly 2 will hit the target is

The letters of the word 'LOGARITHM' are arranged at random. The probability that arrangement starts with vowel and end with consonant is

The p.d.f of c.r.v $$X$$ is given by $$f(x)=\frac{x+2}{18}$$, if $$-2

If the p.m.f of a. r.v. $$X$$ is given by $$P(X=x)=\frac{{ }^5 C_x}{2^5}$$ if $$x=0,1,2, \ldots \ldots . .5=0$$, 0 , otherwise, then which of the foll...

Let $X$ be the number of successes in ' $n$ ' independent Bernoulli trials with probability of success $p=\frac{3}{4}$. The least value of ' $n$ ' so ...

The probability that three cards drawn from a pack of 52 cards, all are red is

$$\begin{aligned} &\text { The pdf of a random variable } X \text { is }\\ &\begin{aligned} f(x) & =3\left(1-2 x^2\right), & & 0 The $P\left(\frac{1}{...

A player tosses 2 fair coins. He wins Rs. 5 if 2 heads appear, Rs. 2 If 1 head appear and Rs. 1 if no head appears, then variance of his winning amoun...

In a bionomial distribution, mean is 18 and variance is 12 then $p=$ ...........

The p.d.f of a random variable $x$ is given by $$\begin{aligned} & f(x)=\frac{1}{4 a}, \quad 00) \\ & =0 \text {, otherwise } \end{aligned}$$ and $P\l...

If three dices are thrown then the probability that the sum of the numbers on their uppermost faces to be atleast 5 is

It is observed that $25 \%$ of the cases related to child labour reported to the police station are solved. If 6 new cases are reported, then the prob...

A bag contains 6 white and 4 black balls. Two balls are drawn at random. The probability that they are of the same colour is ...........

A random variable X has following probability distribution .tg {border-collapse:collapse;border-spacing:0;} .tg td{border-color:black;border-style:s...

If the c.d.f (cumulative distribution function) is given by $F(x)=\frac{x-25}{10}$, then $P(27 \leq x \leq 33)=\ldots \ldots$