Probability Questions (959)

A box contains 2 black, 4 white, and 3 red balls. One ball is drawn at random from the box and kept aside. From the remaining balls in the box, another ball is drawn at random and kept aside the first. This process is repeated till all the balls are drawn from the box. The probability that the balls drawn are in the sequence of 2 black, 4 white, and 3 red is

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7\bar{X}+4\bar{Y}$ is equal to

Let the mean and the standard deviation of the probability distribution $\begin{array}{|c|c|c|c|c|}\hline X & \alpha & 1 & 0 & -3\\\hline P(X) & \frac{1}{3} & K & \frac{1}{6} & \frac{1}{4}\\\hline\end{array}$ be $\mu$ and $\sigma$, respectively. If $\sigma-\mu=2$, then $\sigma+\mu$ is equal to

Two natural numbers x and y are chosen at random. What is the probability that x2 + y2 is divisible by 5?

\(A\) and \(B\) play a game of tennis. The situation of the game is as follows: if one scores two consecutive points after a deuce, he wins; if loss of a point is followed by win of a point, it is deuce. The chance of a server to win a point is 2/3. The game is at deuce and \(A\) is serving. Probability that \(A\) will win the match is (serves are changed after each game)

An artillery target may be either at point I with probability 8/9 or at point II with probability 1/9. We have 55 shells, each of which can be fired either at point I or II. Each shell may hit the target, independent of the other shells, with probability 1/2. Maximum number of shells must be fired at point I to have maximum probability is

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $\gcd(m,n)=1$, then $n-m$ is equal to

A fair coin is tossed \(n\) times. If the probabilities of getting 4 and 6 heads in \(n\) being in AP then \(n\) is equal to

A student can solve 2 out of 4 problems of mathematics, 3 out of 5 problem of physics, and 4 out of 5 problems of chemistry. There are equal number of books of math, physics, and chemistry in his shelf. He selects one book randomly and attempts 10 problems from it. If he solves the first problem, then the probability that he will be able to solve the second problem is

10 balls are thrown into three boxes namely Box 1, Box 2, and Box 3 with respective probabilities 1/4, 1/4 and 1/2. What is the probability that out of 10 balls, 1\(^{\text{st}}\) box will get 2, 2\(^{\text{nd}}\) will get 3 and 3\(^{\text{rd}}\) box will get 5 balls?

A box contains two 50-paise coins, five 25-paise coins and a certain fixed number \(N (\ge 2)\) of 10 and 5-paise coins. Five coins are taken out of the box at random. Find the probability that the total value of these 5 coins is less than 1 rupee and 50 paise.

In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is

Prob detection in n cycles.

If \(a\) and \(b\) are chosen randomly by throwing a pair of fair dice, then the probability that \(\lim\limits_{x \to 0}\left(\dfrac{a^x + b^x}{2}\right)^{\frac{2}{x}} = 6\) equals:

Fourteen numbered balls \((1, 2, 3, \ldots, 14)\) are divided in 3 groups randomly. Find the probability that the sum of the numbers on the balls, in each group, is odd.

Three a's, three b's and three c's are placed randomly in a \(3 \times 3\) matrix. The probability that no row or column contain two identical letters can be expressed as \(\frac{p}{q}\), where p and q are coprime then (p + q) equals to:

The probability that at least one of the events \(A\) and \(B\) occurs is 0.60. If \(A\) and \(B\) occur simultaneously with probability 0.20 then \(P(A') + P(B')\) is equal to

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to

15. The probability that in a year of the 22nd century, chosen at random there will be 53 Sundays, is ___.

Standard overlap problem.

Forty teams play a tournament. Each team plays every other team just once. Each game results in a win for one team. If each team has a 50% chance of winning each game, the probability that at the end of the tournament, every team has won a different number of games is

Given $P(A)=0.5$, $P(A\cup B)=0.8$. If $A$ and $B$ are mutually exclusive, $P(B)=p$. If $A$ and $B$ are independent, $P(B)=q$. Find $q/p$.

Lot $A$ consists of 5 good and 3 defective articles. Lot $B$ consists of 3 good and 5 defective articles. A new lot $C$ is formed by taking 3 articles from $A$ and 4 articles from $B$. The probability that an article chosen at random from $C$ is defective, is:

Two families each having 4 members are to be seated around a circular table with alternate red and blue chairs. If probability that members of same family are seated together is $p$, then $35p$ equals

There are two townships $A$ and $B$ in a city containing 40\% and 60\% of the population respectively. 15\% of the total population suffer from heart disease. $P(\text{heart disease}|A) = 6P(\text{heart disease}|B)$. A person randomly diagnosed turns out to be free from heart disease; then the probability that he lives in township $B$ is

In a single throw of two dice what is the probability of obtaining a number greater than 7, if 4 appears on the first dice?

For Problems 16–18: Two fair dice are rolled. Let \(P(A_i) > 0\) denote the event that the sum of the faces of the dice is divisible by \(i\).The number of all possible ordered pairs \((i, j)\) for which the events \(A_i\) and \(A_j\) are independent is

Let $S=\{2,3,5,7,11\}$. $A$ and $B$ are two matrices of order 2 each with distinct elements from $S$. Probability that matrix $AB$ has at least one odd entry is

If $A$ is any event in a sample space, the maximum value of $3\sqrt{P(A)}+4\sqrt{P(A')}$ is

If $a$ and $b$ are chosen randomly by throwing a pair of fair cubical dice, then the probability that $\displaystyle\lim_{x\to0}\left(\frac{a^x+b^x}{2}\right)^{2/x} = 6$ equals

The probability that a randomly chosen 5-digit number formed from the digits 1, 2, 3, 4, 5 (without repetition) is divisible by 4, given that the number is even, is

There are two townships $A$ and $B$ in a city containing 40\% and 60\% of the population respectively. 15\% of the total population suffer from heart disease. $P(\text{heart disease}|A) = 6P(\text{heart disease}|B)$. A person randomly diagnosed turns out to be free from heart disease; then the probability that he lives in township $B$ is

If 10 different balls are placed in 4 distinct boxes at random, the probability that two of these boxes contain exactly 2 and exactly 3 balls is [JEE Main 2020]

A bag contains 30 white and 10 red balls. 16 balls are drawn with replacement. Let \(X\) = number of white balls drawn. The value of \(\dfrac{\text{mean} + \text{S.D.}}{\text{mean} - \text{S.D.}}\) is [JEE Main 2020]

Two numbers are randomly selected from the set \(\{1, 2, 3, 4, 5, 6\}\). Given that their sum is even, the probability that both numbers are odd is

In a college, 25% of the boys and 10% of the girls offer Mathematics. The girls constitute 60% of the total number of students. If a student is selected at random and is found to be studying Mathematics, the probability that the student is a girl is

In class XI of a school, 40% of the students study Mathematics and 30% study Biology. 10% of the class study both Mathematics and Biology. If a student is selected at random from the class, find the probability that he will be studying Mathematics or Biology.

Let \(A\) and \(B\) be two events such that \(P(A \cup B) = 1/6\), \(P(A \cap B) = 1/4\) and \(P(\bar{A}) = 1/4\), where \(\bar{A}\) stands for complement of event \(A\). Then events \(A\) and \(B\) are

A boy comes from a family of two children. What is the probability that the other child is his sister?

There are two families each having \(n\) children. Tickets are distributed among all the children. The probability that all tickets go to the children of family \(B\) is \(\dfrac{1}{12}\). Find \(n\).

If \(X\) has a binomial distribution, \(B(n, p)\) with parameters \(n\) and \(p\) such that \(P(X = 2) = P(X = 3)\), then \(E(X)\), the mean of variable \(X\), is

A box contains 100 tickets numbered 1, 2, … 100. Two tickets are chosen at random. It is given that the maximum number on the two chosen tickets is not more than 10. The minimum number on them is 5 with probability is

Let $P(A\cap B)=\frac{1}{4}$ and $P(B)=\frac{1}{3}$, where $P\!\left(\dfrac{A\cap B}{A\cup B}\right)=\dfrac{k}{22}$. Then $k$ is

Let $E_1, E_2, E_3$ be three independent events such that $3P(E_1\cap\bar{E_2}\cap\bar{E_3})=P(\bar{E_1}\cap E_2\cap\bar{E_3})=9P(\bar{E_1}\cap\bar{E_2}\cap E_3)=3-3P(E_1\cup E_2\cup E_3)$. If the absolute value of $\begin{vmatrix}P(E_1)&P(E_2)&P(E_3)\\P(E_2)&P(E_3)&P(E_1)\\P(E_3)&P(E_1)&P(E_2)\end{vmatrix}=\dfrac{a}{b}$ where $a,b\in\mathbb{N}$, then least value of $a+b$ is

If \(a\) is an integer lying in \([-5, 30]\), then the probability that the graph of \(y = x^2 + 2(a+4)x - 5a + 64\) is strictly above the \(x\)-axis is

Find the probability of drawing either an ace or a king from a pack of card in a single draw.

If $A$ is any event in a sample space, the maximum value of $3\sqrt{P(A)}+4\sqrt{P(A')}$ is

There are two townships $A$ and $B$ in a city containing 40\% and 60\% of the population respectively. 15\% of the total population suffer from heart disease. $P(\text{heart disease}|A) = 6P(\text{heart disease}|B)$. A person randomly diagnosed turns out to be free from heart disease; then the probability that he lives in township $B$ is

A shooter hits a target with probability $\frac{1}{4}$. She fires until she hits 3 times. Probability that she fires exactly 6 bullets lies in the interval

Two integers \(x\) and \(y\) are chosen with replacement out of the set \(\{0, 1, 2, 3, \ldots, 10\}\). Then find the probability that \(|x - y| > 5\).