Probability Questions (959)

The probability so that $r$ $1 \times 1$ squares which are selected from a $m \times n$ chess board such that no two of them share the same row or same column is :

The probabilities of events $A \cap B$, $A$, $B$ and $A \cup B$ are respectively in A.P. with second term equal to the common difference. Therefore, $A$ and $B$ are:

The probabilities of events $A \cap B$, $A$, $B$ and $A \cup B$ are respectively in A.P. with second term equal to the common difference. Therefore, $A$ and $B$ are:

A student has to match historical events viz., Dandi march, Quit India Movement and Mahatma Gandhi's assassination with the years 1948, 1930 and 1942. The student has no knowledge of the correct answer decides to match the events and years randomly. Let $E_i(0 \leq i \leq 3)$ denote the event that the student gets exactly $i$ correct answers, then which of the following is/are NOT correct?

$A$ 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)

A fair coin is tossed $n$ times. Let $X$ denote the number of times head occurs. If $P(X = 4)$, $P(X = 5)$ and $P(X = 6)$ are in arithmetic progression, then the value of $n$ can be:

One card is missing from a pack of cards. Let $A$ be the event that missing card is a spade. Then two cards are drawn, and $S$ be the event that they are spades then:

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:

One card is missing from a pack of cards. Let $A$ be the event that missing card is a spade. Then two cards are drawn, and $S$ be the event that they are spades then:

Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is :

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

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

If $\frac{1+4p}{4}$, $\frac{1-p}{3}$ and $\frac{1-2p}{2}$ are the probabilities of three mutually exclusive events then $p$ may be:

5 players of equal strength play one each with each other. $P(A) =$ probability that at least one player wins all matches he (they) plays. $P(B) =$ probability that at least one player loses all his (their) matches. Then:

$2n$ 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:

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 :

Three different dice are rolled three times. The Probability that they show different numbers only two times is:

Two persons $A$ and $B$ have respectively $n+1$ and $n$ coins, which they toss simultaneously. Then probability $P$ that $A$ will have more heads then $B$ belongs:

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

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

A square is inscribed in a circle. If $p_1$ is the probability that a randomly chosen point of the circle lies within the square and $p_2$ is the probability that the point lies outside the square, then:

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

If $A$ and $B$ are exhaustive events in a sample space such that probabilities of the events $A \cap B$, $A$, $B$ and $A \cup B$ are in A.P. If $P(A) = K$, where $0 < K \leq 1$, then:

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 :

A boy has a collection of the blue and green marbles. The number of blue marbles belong to the set $\{2, 3, 4, \ldots, 13\}$. If two marbles are chosen simultaneously and random from his collection, then the probability that they have different colours is $1/2$. Possible number of blue marbles is:

Out of 6 pairs of distinct gloves 8 gloves are randomly selected, then the probability that there exist exactly 2 pairs in it is :

A fair coin is tossed 5 times then the probability that no two consecutive heads occur, is:

Players $P_1, P_2, P_3, \ldots, P_n$ of equal skill, play a game consecutively in pairs as $P_1P_2, P_2P_3, P_3P_4, \ldots, P_nP_1, \ldots$ and any player who wins two consecutive games (i.e $k$ and $(k+1)$th game) wins the match. If the chance that the match is won at the $r$th game is $k$ then:

A bag contains 20 blue marbles, 12 red marbles and some other number of green marbles. If the probability of drawing green marble in one try is $\frac{1}{y}$ then which of the following statements is/are correct?

If $A_1, A_2, \ldots, A_n$ be any events of the same sample space then:

$2n$ 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:

Four die are thrown simultaneously. The probability that 4 and 3 appear on two of the die given that 5 and 6 have appeared on other two die is:

A bag contains four tickets marked with numbers 112, 121, 211, and 222. One ticket is drawn at random from the bag. Let $E_i$ (i = 1, 2, 3) denote the event that $i^{th}$ digit on the ticket is 2. Then:

The letters of the word PROBABILITY are written down at random in a row. Let $E_1$ denote the event that two $i$, $s$ are together and $E_2$ denote the event that two $B$'s are together, then:

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 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

Three of six vertices of a regular hexagon are chosen at random. The probability that the triangle with three vertices is equilateral equal to :

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

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:

Players $P_1, P_2, P_3, \ldots, P_n$ of equal skill, play a game consecutively in pairs as $P_1P_2, P_2P_3, P_3P_4, \ldots, P_nP_1, \ldots$ and any player who wins two consecutive games (i.e $k$ and $(k+1)$th game) wins the match. If the chance that the match is won at the $r$th game is $k$ then:

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

A board has 16 squares as shown in the figure (4$\times$4 grid). Out of these 16 squares, two squares are chosen at random. The probability that they have no side in common is:

Let $A = [a_{ij}]$ be a square matrix of order 2 with entries either 0 or 1. Let $E$ be the event that $A$ is an invertible matrix. Then the probability $P(E)$ is:

Two integers are selected at random from \(\{1, 2, 3, \ldots, 11\}\). Given that the sum is even, the conditional probability that both numbers are odd is [JEE Main 2019]

A cube painted red on all sides is cut into 125 equal small cubes. A small cube when picked is found to show red color on one of its faces. The probability that two more faces also show red color is :

Total cases = $^nC_2$. The number of favorable cases when $j$ is placed in pth of a is $^nC_1 × C_{j-1}^1$. Find required probability.

Let M be the maximum value of the product of two positive integers when their sum is 66. Let the sample space $S = \{x \in \mathbb{Z} : x(66-x) \geq \frac{5}{9}M\}$ and the event $A = \{x \in S : x \text{ is a multiple of } 3\}$. Then P(A) is equal to

Two dice A and B rolled. Variance of $\alpha-\beta$ is $\dfrac{p}{q}$ (coprime). Sum of positive divisors of $p$ is equal to

A bag has 6 white and 4 black balls. A die is rolled; $n$ balls are drawn equal to the die's number. $P$(all drawn balls are white) is