Probability Questions (959)

If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^\text{th}$ roll than the number obtained in the $(i-1)^\text{th}$ roll, $i=2,3$, is equal to

In a tournament, a team plays 10 matches with probabilities of winning and losing each match as $\frac{1}{3}$ and $\frac{2}{3}$ respectively. Let $x$ be the number of matches that the team wins, and $y$ be the number of matches that team loses. If the probability $P(|x-y|\leq2)$ is $p$, then $3^9p$ equals

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

Two marbles are drawn in succession from a box containing 10 red, 30 white, 20 blue and 15 orange marbles, with replacement being made after each drawing. Then the probability that first drawn marble is red and second drawn marble is white, is

A card from a pack of 52 cards is lost. From the remaining 51 cards, n cards are drawn and are found to be spades. If the probability of the lost card to be a spade is 11 50 , the n is equal to

If the probability that the random variable X takes the value x is given by P$(X = x) = k(x + 1)3 -x$,$x = 0$, 1, 2, 3$\ldots$$\ldots$, where k is a constant, then P(X$\ge$3) is equal to

A coin is tossed three times. Let $X$ denote the number of times a tail follows a head. If $\mu$ and $\sigma^{2}$ denote the mean and variance of $X$, then the value of $64(\mu+\sigma^{2})$ is:

Let $A=[a_{ij}]$ be a $2\times 2$ matrix such that $a_{ij}\in\{0,1\}$ for all $i,j$ and $P(a_{ij}=0)=P(a_{ij}=1)=\dfrac{1}{2}$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then the variance of $X$ is:

Bag $1$ contains $4$ white balls and $5$ black balls, and Bag $2$ contains $n$ white balls and $3$ black balls. One ball is drawn randomly from Bag $1$ and transferred to Bag $2$. A ball is then drawn randomly from Bag $2$. If the probability that the ball drawn is white is $\dfrac{29}{45}$, then $n$ is equal to:

Bag $B_{1}$ contains $6$ white and $4$ blue balls, Bag $B_{2}$ contains $4$ white and $6$ blue balls, and Bag $B_{3}$ contains $5$ white and $5$ blue balls. One of the bags is selected at random and a ball is drawn from it. If the ball is white, then the probability that the ball is drawn from Bag $B_{2}$ is:

Two balls are selected at random one by one without replacement from a bag containing $4$ white and $6$ black balls. If the probability that the first selected ball is black, given that the second selected ball is also black, is $\dfrac{m}{n}$ where $\gcd(m,n)=1$, then $m+n$ is equal to:

Let $S$ be the set of all the words that can be formed by arranging all the letters of the word GARDEN. From the set $S$, one word is selected at random. The probability that the selected word will NOT have vowels in alphabetical order is:

$A$ and $B$ are two events such that $P(A\cap B)=0.1$, and $P(A\mid B)$ and $P(B\mid A)$ are the roots of the equation $12x^{2}-7x+1=0$, then the value of $\dfrac{P(\bar{A}\cup\bar{B})}{P(\bar{A}\cap\bar{B})}$ is:

Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $X$ denotes the number of defective oranges, then the variance of $X$ is:

For any two events \(A\) and \(B\),

In a lottery all the tickets are blank except one, on which there is a prize. \(n\) persons draw a ticket each one after another without replacement. The probabilities of the 6th person to win the prize is \(\frac{6}{n}\).

Four persons hit a target with respective probabilities \(\dfrac{1}{2},\dfrac{1}{3},\dfrac{1}{4},\dfrac{1}{8}\) independently. The probability that the target is hit by exactly one person is [JEE Main 2019]

An urn contains 6 white and 9 black balls. Two successive draws of 4 balls are made without replacement. The probability that the first draw gives all white balls and the second draw gives all black balls is:

A fair die is thrown until 2 appears. Then the probability that 2 appears in even number of throws is:

An integer is chosen at random from the integers $1,2,3,\ldots,50$. The probability that the chosen integer is a multiple of at least one of $4$, $6$ and $7$ is

Let A be the event that the absolute difference between two randomly chosen real numbers in the sample space $[0, 60]$ is less than or equal to $a$. If $P(A) = \frac{11}{36}$, then $a$ is equal to ___.

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3)$, $b=P(X\geq3)$ and $c=P(X\geq6\mid X>3)$. Then $\dfrac{b+c}{a}$ is equal to

Let Ajay will not appear in JEE exam with probability $p=\dfrac{2}{7}$, while both Ajay and Vijay will appear in the exam with probability $q=\dfrac{1}{5}$. Then the probability that Ajay will appear in the exam and Vijay will not appear is:

The probability of the two events are 0.25 and 0.50. The probability of both happening together is 0.14. Which of the following is the probability of none of the events happening?

A consignment of 15 radios contains 4 defectives. The radios are taken out one by one at random and examined. The ones examined are not put back. What is the probability that the ninth one examined is the last defective?

Two numbers $k_1$ and $k_2$ are randomly chosen from the set of natural numbers. Then, the probability that the value of $i^{k_1} + i^{k_2}$, $(i = \sqrt{-1})$ is non-zero, equals:

$A$ and $B$ alternately throw a pair of dice. $A$ wins if he throws a sum of 5 before $B$ throws a sum of 8, and $B$ wins if he throws a sum of 8 before $A$ throws a sum of 5. The probability that $A$ wins if $A$ makes the first throw, is:

Three defective oranges are accidentally mixed with seven good ones and on looking at them, it is not possible to differentiate between them. Two oranges are drawn at random from the lot. If $x$ denotes the number of defective oranges, then the variance of $x$ is

Let $A = [a_{ij}]$ be a $2 \times 2$ matrix such that $a_{ij} \in \{0, 1\}$ for all $i$ and $j$. Let the random variable $X$ denote the possible values of the determinant of the matrix $A$. Then, the variance of $X$ is:

If three letters can be posted to any one of the 5 different addresses, then the probability that the three letters are posted to exactly two addresses is:

Two dice are thrown. The probability that the number appeared have a sum of 8. If it is known that the second die always exhibits 4, is

18. Two persons X and Y threw two dice each. The probability that Y throws a sum greater than X is when it is known that X throws a sum of at least 8, is ___.

A lot contains 50 defective and 50 non-defective bolts. Two bolts are drawn successively without replacement. The probability that the second bolt is defective given that the first is non-defective is

One die has two faces marked $1$, two faces marked $2$, one face marked $3$ and one face marked $4$. Another die has one face marked $1$, two faces marked $2$, two faces marked $3$ and one face marked $4$. The probability of getting the sum of numbers to be $4$ or $5$, when both the dice are thrown together, is:

A bag contains 19 unbiased coins and one coin with head$o_n$both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was unbiased, is , then$n - m$is equal to m 2 2 , gcd(m,$n) = 1$n :

A box contains 10 pens of which 3 are defective. A sample of 2 pens is drawn at random and let X denote the number of defective pens. Then the variance of X is

Two red, three green and four blue balls are placed in a row at random. The probability that the two red balls are not adjacent is

17. There are two coupons each with a word written on them. On one of them the word PATNA is written and on the other PLATE. A coupon is taken at random and 3 letters are selected at random from the letters of the word on the coupon. The probability that the selection contains two vowels, is ___.

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

A speaks truth 3 times out of 4 while B, 7 times out of 10. A ball is drawn at random from a bag containing one black ball and five other balls of different colours. Both A and B report that a black ball has been drawn from the bag. Find the probability of their assertion being true?

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

Mr. A lives at origin on the Cartesian plane and has his office at (4, 5). His friend lives at (2, 3) on the same plane. Mr. A can go to his office traveling one block at a time either in the +y or +x direction. If all possible paths are equally likely then the probability that Mr. A passed his friends house is (shortest path for any event must be considered)

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

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

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

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

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 card is drawn and replaced in an ordinary pack of 52 cards. The minimum number of times a card must be drawn so that the probability of getting at least one ace exceeds \(\dfrac{1}{2}\) is

A bag contains 4 yellow and 8 white balls (12 total). Two balls are drawn at random without replacement. The probability that both drawn balls are white 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: