Probability Questions (959)

Mr. A forgot a phone number. He remembers it starts with 713 and the remaining 4 digits are some arrangement of 1, 7, 5, 9 (each used exactly once). He dials a number at random. The probability that he dials the correct number is

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is

A coin is tossed and a die is thrown simultaneously. The probability that the outcome is a head or a number greater than 4 on the die is

There are three bugs each containing $5$ white balls and $2$ black balls and $2$ bags each containing $1$ white ball and $4$ black balls; a black ball having been drawn the probability that it came from the first group is ______.

Bag A contains 9 white and 8 black balls, while bag B contains 6 white and 4 black balls. One ball is randomly picked from bag B and mixed up with the balls in bag A. Then a ball is randomly drawn from bag A. If the probability that the ball drawn is white is $\dfrac{p}{q}$, $\gcd(p,q)=1$, then $p+q$ is equal to

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x + y + z = 1$, $2x + Ny + 2z = 2$, $3x + 3y + Nz = 3$ has unique solution is $\frac{k}{6}$, then the sum of value of k and all possible values of N is

The probability that two queens, placed at random on a chess board, do not take on each other is $\frac{p}{q}$ (where $H.C.F(p, q) = 1$) then $q - p - 5 = \ldots\ldots\ldots\ldots$

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

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

From a lot containing 10 defective and 90 non-defective bulbs, 8 bulbs are selected one by one with replacement. Then the probability of getting at least 7 defective bulbs is

Let the probability of getting any number other than 5 is P, then the probability of getting 5 is 5P. $P + P + P + P + 5P = 1$, P = $\frac{1}{9}$. Expected income per throw = $x imes (\frac{1}{9} + 8)$.

Let $\Omega$ be the sample space and $A \subseteq \Omega$ be an event. Given below are two statements: (S1): If $P(A) = 0$, then $A = \phi$. (S2): If $P(A) = 1$, then $A = \Omega$. Then

Six different balls are put in three different boxes, no box being empty. The probability of putting balls in the boxes in equal numbers is :

$2n$ balls (all distinct in size) are arranged in a row. First few of these balls are black rest all white, both odd in number. The probability that there is exactly, one black ball in one of all possible arrangements is:

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is

Total cases = $^nC_2$. For favorable cases, let $a = 9$, $b = 9$. Find $i$ such that $|i - j| ≤ 3$ (must be an integer).

A point $X$ is selected at random from a line segment $AB$ with mid point $O$. The probability that the line segments $AX, XB$ and $AO$ can form a triangle is :

The probability that a positive two-digit number selected at random has its tens digit at least 3 more than its units digit is

A coin of diameter $1/2$ is tossed randomly onto the rectangular cartesian plane. The probability that the coin does not intersect any line whose equation is of the form $x = k$, or $y = k,k$ is integer, is:

A point is selected at random inside an equilateral triangle whose side length is 3. The probability its distance to any corner is greater than 1 is

A student appears for test I, II and III. The student is successful if he passes either in test I, II or I, III. The probability of the student passing in test I, II and III are respectively $p$, $q$ and $1/2$. If the probability of the student to be successful is $1/2$ then :

A point is selected at random inside a circle. The probability that the point is closer to the centre of the circle than to its circumference :

A committee of 3 persons is to be randomly selected from a group of 3 men and 2 women, and the chair person will be selected from the committee. The probability that the committee will have exactly 2 women and 1 man, and that the chairperson will be a woman, is

If an unbiased die, marked with $-2, -1, 0, 1, 2, 3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is:

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\dfrac{k}{3^{11}}$, then $k$ is equal to

If \(a\), \(b\), \(c\) are three numbers selected at random without repetition from the set \(\{-1, 0, 1, 2, 3\}\), the probability that \(a + b + c = 0\) is

NTA Test 18 (Single Choice) A small pack of cards consists of 5 green cards, 4 blue cards and 3 black cards. The pack is shuffled through and first three cards are turned face up. The probability that there is exactly one card of each colour is

Let $A$ and $B$ be two events such that $P\left(A \cap B^{c}\right) = 0.20, P\left(A^{c} \cap B\right) = 0.15, P\left(A^{c} \cap B^{c}\right) = 0.1$, then $p(A/B)$ is equal to,

A license plate consists of 2 English alphabet letters followed by 4 digits (each chosen independently and uniformly at random). The probability that the license plate reads the same forwards and backwards (palindrome) is

In a binomial distribution $B(n, p)$, the sum and product of the mean and variance are 5 and 6 respectively, then find $6(n + p - q)$ is equal to:

Fifteen coupons are numbered $1, 2, 3, \ldots, 15$ respectively. Seven coupons are selected at random one at a time with replacement. The probability, that the largest number appearing on selected coupons is at most $B$, is

Three smallest squares are chosen randomly on a chess board are the probability that these squares have exactly two corners, but no side common is:

An experiment yields 3 mutually exclusive and exhaustive events $A$, $B$ and $C$. If $P(A) = 2P(B)$ and $2P(C) = 3P(B)$, then $P(A)$ is equal to

A machine containing $n$ different balls, when switched on, can throw up any number of balls one by one. The probability of throwing $r$ balls is directly proportional to $r$. Given that a particular ball is the first ball to pop up, the probability that machine has thrown up all the balls is:

Two subsets $A$ and $B$ of a set containing $n$ elements are chosen at random. The probability that $A \subseteq B$ is:

Each of 10 passengers board any of the three buses randomly which had no passenger initially. The probability that each bus has got at least one passenger is :

If a random variable $x$ has the probability distribution $P(x)$: $0,2k,k,3k,2k^2,2k,k^2+k,7k^2$ for $x=0,1,2,3,4,5,6,7$, then $P(3<x\leq6)$ is equal to

The probability that a randomly chosen 3 digit number has exactly 3 factors :

A box contains a red balls and $12$ black balls. $3$ balls are drawn one by one without replacement. If the probability of choosing $3$ red balls is equal to the probability of choosing $2$ red and $1$ black ball, then the possible value of $a$ can be

An urn contains 5 red and 2 green balls. A ball is drawn at random. If green, it is replaced by a red ball in the urn. Another ball is then drawn. The probability that the second ball is red is [JEE Main 2019]

If two of the 64 squares on a chessboard are chosen at random, the probability that they have a side in common is

NTA Test 14 (Single Choice) For an admission test of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate can solve any problem is $\frac{1}{3}$, then the probability that he is unable to solve less than two problems is

$8n$ players $P_1, P_2, \ldots, P_{8n}$ play a knock out tournament. It is known that all the players are of equal strength. The tournament is held in 3 rounds where the players are paired at random in each round. If it is given that $P_1$ wins in the third round. The probability that $P_2$ looses in the second round is:

A number is chosen at random from the numbers 10 to 99. By seeing the number a man will laugh if product of the digits is 12. If he choose three numbers with replacement then the probability that he will laugh at least once is:

A determinant is chosen at random from the set of all \(2 \times 2\) determinants with entries from \(\{0, 1\}\). The probability that the value of the chosen determinant is positive is

In a test, an examinee either guesses or knows the answer to a multiple choice question with four choices. The probability that he makes a guess is $\frac{1}{4}$ and the probability that his answer is correct given that he guesses is $\frac{1}{4}$. The probability that his answer is correct given that he knows is $1$. The probability that he knew the answer to the question given that he correctly answered, is

The probability that in a random arrangement of the word MATHEMATICS, the two M's are not together given that the two A's are not together is

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

Two positive real numbers $x$ and $y$ satisfying $x\leq1$ and $y\leq1$ are chosen at random. The probability that $x+y\leq1$, given that $x^2+y^2\geq\frac{1}{4}$, 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