Probability practice · Mixed difficulty

Four identical dice are rolled once. The probability that atleast three different numbers appear on them, is

One hundred cards are numbered from 1 to 100. The probability that a randomly chosen card has a digit 5 is

If the integers $m$ and $n$ are chosen at random between 1 and 100, then the probability that a number of the form ${7^m} + {7^n}$ is divisible by 5, equals

The value of $C$ for which $P\left( {X = k} \right) = C\;{k^2}$ can serve as the probability function of a random variable $X$ that takes value 0, 1, 2, 3, 4 is

Two dice are thrown independently. Let $A$ be the event that the number appeared on the $1^{\text {st }}$ die is less than the number appeared on the $2^{\text {nd }}$ die, $B$ be the event that the number appeared on the $1^{\text {st }}$ die is even and that one the second die is odd, and $C$ be the event that the number appeared on the $1^{\text {st }}$ die is odd and that on the $2^{\text {nd }}$ is even. Then

A die is tossed 5 times. Getting an odd number is considered a success. Then, the variance of distribution of success, is

Three randomly chosen non negative integers $x, y$ and $z$ are found to satisfy eq. $x+y+z=10$ and the probability that $z$ is even is $p / q$ ($p, q$ coprime) then ${p}+{q}=$ ____

A group of $2 n$ boys and $2 n$ girls is randomly divided into two equal groups. The probability that each group contains the same number of boys and girls is

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to

Given $P\left( {A \cup B} \right) = 0.6,P\left( {A \cap B} \right) = 0.2,$ the probability of exactly one of the event occurs is

A biased die is marked with numbers 2, 4, 8, 16, 32, 32 on its faces and the probability of getting a face with mark n is 1/n. If the die is thrown thrice, then the probability, that the sum of the numbers obtained is 48, is:

The probability that a certain kind of component will survive a given shock test is $\frac{3}{4}$. The probability that exactly 2 of the next 4 components tested survive is

Let $S=\{1,2,3, \ldots, 2022\}$. Then the probability, that a randomly chosen number $n$ from the set $S$ such that $\operatorname{HCF}(n, 2022)=1$, is

A student takes his examination in four subjects $\alpha, \beta$, $\gamma, \delta$. He estimates his chance of passing in $\alpha$ as $\frac{4}{5}$, in $\beta$ as $\frac{3}{4}$, in $\gamma$ as $\frac{5}{6}$ and in $\delta$ as $\frac{2}{3}$. To qualify he must pass in $\alpha$ and at least two other subjects. The Probability that he qualifies is

Three six-faced fair dice are thrown together. The probability that the sum of the numbers appearing on the dice is $k(3 \leq k \leq 8)$, is

Four positive integers are taken at random and are multiplied together. Then the probability that the product ends in an odd digit other than 5, is

If $m$ rupee coins and $n$ ten paise coins are placed in a line, then the probability that the extreme coins are ten paise coins, is

Two cards are drawn successively with replacement from a well shuffled deck of 52 cards, then the mean of the number of aces is

Let ${E_1},{E_2}$ be two mutually exclusive events of an experiment with $P$(not ${E_2}$) $= 0.6 = P\left( {{E_1} \cup {E_2}} \right)$.Then, $P\left( {{E_1}} \right)$ is equal to

Four boys and three girls stand in a queue for an interview, probability that they will in alternate position, is

A missile is destroyed before hitting the target by a probability of $\frac{1}{3}$. If the missile is not destroyed probability of hitting the target is $\frac{3}{4}$. Three missiles are fired. The probability of exactly two hits the target is

A four digit number is formed by the digits 1, 2, 3, 4 with no repetition. The probability that the number is odd, is

Assertion: Probability of getting a head in a toss of an unbiased coin is $\frac{1}{2}$.

Reason: In a simultaneous toss of two coins, the probability of getting 'no tails' is $\frac{1}{4}$.

$A$ and $B$ are two independent events. The probability that both $A$ and $B$ occur is $1/6$ and the probability that neither of them occurs is $1/3$. Then,

$A$ is a set containing $n$ elements. Two subsets $P$ and $Q$ of $A$ are chosen at random. ($P$ and $Q$ may have elements in common). The probability that $P \cap Q=$ $\phi$ is

A bag contains 16 coins of which two are counterfeit with heads on both sides. The rest are fair coins. One is selected at random from the bag and tossed. The probability of getting a head is

If $A$ and $B$ are independent events of $a$ random experiment such that $P(A \cap B)=\frac{1}{6}$ and $P(\bar{A} \cap \bar{B})=\frac{1}{3}$, then $P(A)$ is equal to

A bag $X$ contains 2 white and 3 black balls and another bag $Y$ contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, probability for the ball chosen be white, is

On each evening a boy either watches DOORDARSHAN channel or TENSPORTS. The probability that he watches TEN SPORTS is $\frac{4}{5}$. If he watches DOORDARSHAN, there is a chance of $\frac{3}{4}$ that he will be asleep, while it is $\frac{1}{4}$ when he watches TEN SPORTS. On one day, the boy is found to be asleep. The probability that the boy watched DOORDARSHAN is

Five different objects ${A_1},\,{A_2},\,{A_3},\,{A_4},\,{A_5}$ are distributed randomly in 5 places marked 1, 2, 3, 4, 5. One arrangement is picked at random. The probability that in the selected arrangement, none of the object occupies the place corresponding to its number, is