Probability Questions (959)

Three married couples sit in a row. Find the probability that no husband sits with his wife.

For Problems 4–6: In an objective paper, there are two sections of 10 questions each. For 'section 1', each question has 5 options and only one option is correct and 'section 2' has 4 options with multiple answers and marks for a question in this section is awarded only if he ticks all correct answers. Marks for each question in 'section 1' is 1 and in 'section 2' is 3. (There is no negative marking.)If a candidate in total attempts 4 questions all by guessing, then the probability of scoring 10 marks is

A, B, C are events such that \(P(A) = 0.3\), \(P(B) = 0.4\), \(P(C) = 0.8\), \(P(AB) = 0.08\), \(P(AC) = 0.28\) and \(P(ABC) = 0.09\). If \(P(A \cup B \cup C) \geq 0.75\), then show that \(P(BC)\) lies in the interval \(0.23 \leq x \leq 0.48\).

An urn contains five balls. Two balls are drawn and are found to be white. Find the probability that all the balls are white.

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

Four numbers are chosen from \(\{1, 2, \ldots, 20\}\). What is the probability that the chosen numbers are in arithmetic progression (AP)?

\(A\) and \(B\) are two independent events such that \(P(A) = 0.3\) and \(P(A \cup B) = 0.8\). Then

One ticket is selected at random from 100 tickets numbered 00, 01, 02, ..., 98, 99. If \(x_1\) and \(x_2\) denotes the sum and product of the digits on the tickets, then \(P(x_1 = 9/x_2 = 0)\) is equal to

A bag contains \(n\) white and \(n\) black balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each pair consists of one white and one black ball is

In a bag there are 6 balls of which 3 are white and 3 are black. They are drawn successively with replacement. What is the chance that the colours are alternate?

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

If \(A\) and \(B\) each toss three coins. The probability that both get the same number of heads is

The probability that a student is not a swimmer is 1/5. Then find the probability that out of 5 students exactly 4 are swimmers.

A box contains N coins, m of which are fair and the rest are biased. The probability of getting a head when a fair coin is tossed is 1/2, while it is 2/3 when a biased coin is tossed. A coin is drawn from the box at random and is tossed twice. The first time it shows head and the second time it shows tail. What is the probability that the coin drawn is fair?

Let \(P(A)\), \(P(B)\) and \(P(C)\) denote the probability of solving a problem by \(A\), \(B\) and \(C\) respectively, where \(P(A) = \frac{1}{2}\), \(P(B) = \frac{1}{3}\) and \(P(C) = \frac{1}{4}\). The probability that the problem is solved is:

A bag contains 10 balls: 4 red and 6 blue. Two balls are drawn without replacement. The probability that both are red is

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:

A fair coin is tossed n times. If the probability that head occurs 6 times is equal to the probability that head occurs 8 times, then find the value of n.

Let \(A\) and \(B\) be two events. Suppose \(P(A) = 0.4\), \(P(B) = p\), and \(P(A \cup B) = 0.7\). The value of \(p\) for which \(A\) and \(B\) are independent is

Four persons independently solve a certain problem correctly with probabilities \(\dfrac{1}{2}, \dfrac{3}{4}, \dfrac{1}{4}, \dfrac{1}{8}\). Then the probability that the problem is solved correctly by at least one of them is

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

If \(C\) and \(D\) are two events such that \(C \subseteq D\) and \(P(D) \neq 0\), then the correct statement among the following is

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

Let \(X\) be a set containing 10 elements and \(P(X)\) be its power set. If \(A\) and \(B\) are picked up at random from \(P(X)\), with replacement, then the probability that \(A\) and \(B\) have equal number of elements, is

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 two additional balls of the same colour are returned to the bag. If now a ball is drawn at random from the bag, then the probability that this drawn ball is red, is

A pair of four dice is thrown independently three times. The probability of getting a score of exactly 9 twice is

The minimum number of times a fair coin needs to be tossed, so that the probability of getting at least two heads is at least 0.96, is ______. (JEE Advanced 2015)

Matrices of order \(2 \times 2\) are formed by using the elements of the set \(A = \{-2, -1, 0, 1, 2\}\), then probability that matrix is either symmetric or skew-symmetric, is greater than:

Consider a sample space "S" representing the adults in a small town who have completed the requirements for a college degree. They have been categorized according to sex and employment as follows:EmployedUnemployedMale46040Female140260An employed person is selected at random. Find the probability that the chosen one is a male.

A fair coin is tossed 100 times. The probability of getting tails 1, 3, ..., 49 times is

The probability that two randomly selected subsets of the set \({1, 2, 3, 4, 5}\) have exactly two elements in their intersection, is

The probability of India winning a test match against West Indies is 1/2. Assuming independence from match to match, find the probability that in a match series India's second win occurs at the third test.

The mean and variance of a random variable having a binomial distribution are 4 and 2, respectively, then \(P(X=1)\) is

In a game A throws two ordinary dice. If he throws 7 or 11 he wins. If he throws 2, 3 or 12 he loses. If he throws any other number, he throws again and continues to throw until either the number he threw first or 7 turns up. In the first case he wins and in the second he loses. Show that the odds against his winning is 251 : 244.

Let A be a set containing n elements. A subset P of the set A is chosen at random. The set A is reconstructed by replacing the elements of P, and another subset Q of A is chosen at random. The probability that \(P \cap Q\) contains exactly m (m < n) elements is

Find the probability that the birthdays of six different persons will fall in exactly two calendar months.

India plays two matches each with West Indies and Australia. In any match the probabilities of India getting points 0, 1 and 2 are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is __________ (up to four decimal places).

Six points are there on a circle from which two triangles are drawn with no vertex common. Find the probability that none of the sides of the triangles intersect.

For Problems 1–3: In a class of 10 students, probability of exactly i students passing an examination is directly proportional to i2. Then answer the following questions:If a student is selected at random, then the probability that he has passed the examination is

If two different numbers are taken from the set \(\{0, 1, 2, 3, \ldots, 10\}\), then the probability that their sum as well as absolute difference are both multiples of 4, is

If \(P(A \cap B) = \dfrac{1}{2}\), \(P(\bar{A} \cap \bar{B}) = \dfrac{1}{3}\), \(P(A) = p\), \(P(B) = 2p\), then find the value of \(p\).

A man and a woman appear in an interview for two vacancies in the same post. The probability of man's selection is 1/4 and that of the woman's selection is 1/3. What is the probability that none of them will be selected?

A hat contains a number of cards with 30% white on both sides, 50% black on one side and white on the other side, 20% black on both sides. The cards are mixed up, and a single card is drawn at random and placed on the table. Its upper side shows up black. The probability that its other side is also black is

Of the three independent events \(E_1\), \(E_2\), and \(E_3\), the probability that only \(E_1\) occurs is \(\alpha\), only \(E_2\) occurs is \(\beta\), and only \(E_3\) occurs is \(\gamma\). Let the probability \(p\) that none of the events \(E_1\), \(E_2\), or \(E_3\) occurs satisfy the equations \((\alpha - 2\beta)\,p = \alpha\beta\) and \((\beta - 3\gamma)\,p = 2\beta\gamma\). All the given probabilities are assumed to lie in the interval \((0, 1)\). Then \[\frac{\text{Probability of occurrence of } E_1}{\text{Probability of occurrence of } E_3} = \underline{\hspace{2cm}}.\] (JEE Advanced 2013)

Two fair dice are thrown till outcome is 12. What is the probability that one has to do 20 throws for this?

In a bag there are six balls of unknown colours, three balls are drawn at random and found to be all black. Find the probability that no black ball is left in the bag. (Or, Find the probability that the bag contained exactly 3 black balls.)

A person goes to office either by car, scooter, bus or train, the probability of which being 1/7, 3/7, 2/7 and 1/7, respectively. Probability that he reaches office late, if he takes car, scooter, bus or train is 2/9, 1/9, 4/9 and 1/9 respectively. Given that he reached office in time, what is the probability that he traveled by a car?

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 __________.

If \(p\) is the probability that a man aged \(x\) will die in a year, then the probability that out of \(n\) men \(A_1, A_2, \ldots, A_n\) each aged \(x\), \(A_1\) will die in an year and be the first to die is

A and B play a game where each is asked to select a number from 1 to 25. If the two numbers match, both of them win a prize. The probability that they will not win a prize in a single trial, is