Probability · JEE Main previous year questions

$A$ 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

1. A.$\frac{8}{19}$
2. B.$\frac{9}{19}$
3. C.$\frac{8}{17}$
4. D.$\frac{9}{17}$

1. A.$\frac{3}{5}$
2. B.$\frac{4}{5}$
3. C.$\frac{23}{30}$
4. D.$\frac{7}{10}$

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 $\frac{m}{n}$, where $\gcd (m,n)=1$, then $m+n$ is equal to :

1. A.4
2. B.14
3. C.11
4. D.13

Let $a$, $b$ and $c$ denote the outcome of three independent rolls of a fair tetrahedral die, whose four faces are marked $1,2,3,4$. If the probability that $a{x}^2+bx+c=0$ has all real roots is $\frac{m}{n}$, $\gcd (m,n)=1$, then $m+n$ is equal to ______.

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

1. A.$5/54$
2. B.$2/54$
3. C.$1/54$
4. D.$3/54$

Let the sum of two positive integers be 24 . If the probability, that their product is not less than $\frac{3}{4}$ times their greatest possible product, is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $n-m$ equals

1. A.$10$
2. B.$11$
3. C.$9$
4. D.$8$

Three balls are drawn at random from a bag containing 5 blue and 4 yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and yellow balls. If $\bar{X}$ and $\bar{Y}$ are the means of $X$ and $Y$ respectively, then $7\bar{X}+4\bar{Y}$ is equal to ________.

There are three bags $X,Y$ and $Z$. Bag $X$ contains 5 one-rupee coins and 4 five-rupee coins; Bag $Y$ contains 4 one-rupee coins and 5 five-rupee coins and Bag $Z$ contains 3 one-rupee coins and 6 five-rupee coins. A bag is selected at random and a coin drawn from it at random is found to be a one-rupee coin. Then the probability, that it came from bag Y, is :

1. A.$\frac{1}{2}$
2. B.$\frac{1}{3}$
3. C.$\frac{5}{12}$
4. D.$\frac{1}{4}$

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 :

1. A.$\frac{18}{25}$
2. B.$\frac{12}{25}$
3. C.$\frac{6}{25}$
4. D.$\frac{4}{25}$

From a lot of 12 items containing 3 defectives, a sample of 5 items is drawn at random. Let the random variable $X$ denote the number of defective items in the sample. Let items in the sample be drawn one by one without replacement. If variance of $X$ is $\frac{m}{n}$, where $\gcd (m,n)=1$, then $n-m$ is equal to ____.

Three urns A, B and C contain 7 red, 5 black; 5 red, 7 black and 6 red, 6 black balls, respectively. One of the urn is selected at random and a ball is drawn from it. If the ball drawn is black, then the probability that it is drawn from urn A is :

1. A.$\frac{4}{17}$
2. B.$\frac{5}{16}$
3. C.$\frac{5}{18}$
4. D.$\frac{7}{18}$

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 $P(|x-y|\le 2)$ is $p$, then $3^{9}p$ equals _________.

A bag contains 8 balls, whose colours are either white or black. 4 balls are drawn at random without replacement and it was found that 2 balls are white and other 2 balls are black. The probability that the bag contains equal number of white and black balls is :

1. A.$\frac{2}{5}$
2. B.$\frac{2}{7}$
3. C.$\frac{1}{7}$
4. D.$\frac{1}{5}$

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

1. A.$\frac{9}{35}$
2. B.$\frac{3}{35}$
3. C.$\frac{24}{35}$
4. D.$\frac{18}{35}$

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

1. A.$\frac{4}{25}$
2. B.$\frac{2}{3}$
3. C.$\frac{2}{25}$
4. D.$\frac{4}{75}$

A coin is biased so that a head is twice as likely to occur as a tail. If the coin is tossed 3 times, then the probability of getting two tails and one head is

1. A.$\frac{1}{9}$
2. B.$\frac{2}{9}$
3. C.$\frac{1}{27}$
4. D.$\frac{2}{27}$

Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,3,\dots ,10\}$. Then the probability that $|x-y|>5$, is :

1. A.$\frac{31}{121}$
2. B.$\frac{60}{121}$
3. C.$\frac{62}{121}$
4. D.$\frac{30}{121}$

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

1. A.$\frac{5}{11}$
2. B.$\frac{5}{6}$
3. C.$\frac{1}{6}$
4. D.$\frac{6}{11}$

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

1. A.$\frac{8}{25}$
2. B.$\frac{9}{50}$
3. C.$\frac{14}{25}$
4. D.$\frac{21}{50}$

A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3)$, $b=P(X\ge 3)$ and $c=P(X\ge 6\mid X>3)$. Then $\frac{b+c}{a}$ is equal to ________.

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 :

1. A.$\frac{3}{256}$
2. B.$\frac{5}{256}$
3. C.$\frac{3}{715}$
4. D.$\frac{5}{715}$

An organization awarded 48 medals in event 'A', 25 in event 'B' and 18 in event 'C'. If these medals went to total 60 men and only five men got medals in all the three events, then, how many received medals in exactly two of three events?

1. A.10
2. B.15
3. C.21
4. D.9

Let the probability of getting head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $\mathrm{N}$ be the number of tosses required. If the probability that the equation $64x^2+5\mathrm{N}x+1=0$ has no real root is $\frac{p}{q}$, where $p$ and $q$ are coprime, then $q-p$ is equal to _________.

Let N denote the sum of the numbers obtained when two dice are rolled. If the probability that $2^N<N!$ is $\frac{m}{n}$, where m and n are coprime, then $4m-3n$ is equal to :

1. A.12
2. B.6
3. C.8
4. D.10

Some couples participated in a mixed doubles badminton tournament. If the number of matches played, so that no couple played in a match, is 840, then the total number of persons, who participated in the tournament, is ___________.