Probability Questions (959)

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 \(|x - y| > 5\).

If \(A\) and \(B\) are two independent events, the probability that both \(A\) and \(B\) occur is \(\frac{1}{8}\) and the probability that neither of them occurs is \(\frac{3}{8}\). Find the probability of the occurrence of \(A\).

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 $q/p$.

If \(P(A) = 0.8\), \(P(B) = 0.5\), and \(P(B/A) = 0.4\), find \(P(A \cup B)\).

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

In a binomial distribution \(B\!\left(b,\, p = \dfrac{1}{4}\right)\), if the probability of at least one success is greater than or equal to \(\dfrac{9}{10}\), then \(n\) is greater than

A signal which can be green or red with probability \(\dfrac{2}{3}\) and \(\dfrac{1}{5}\) respectively, is received by station \(A\) and then transmitted to station \(B\). The probability of each station receiving the signal correctly is \(\dfrac{3}{4}\). If the signal received at station \(B\) is green, then the probability that the original signal was green is

Two cards are drawn one by one randomly from a pack of 52 cards. Then find the probability that both of them are king.

All the jacks, queens, kings, and aces of a regular 52 cards deck are taken out. The 16 cards are thoroughly shuffled and my opponent, a person who always tells the truth, simultaneously draws two cards at random and says, "I hold at least one ace". The probability that he holds two aces is

Three ships \(A\), \(B\), and \(C\) sail from England to India. If the ratio of their arriving safely are 2:5, 3:7, and 6:11, respectively, then the probability of all the ships for arriving safely is