Probability · JEE Main previous year questions

Two different families A and B are blessed with equal number of children. There are 3 tickets to be distributed amongst the children of these families so that no child gets more than one ticket. If the probability that all the tickets go to the children of the family B is $\frac{1}{12}$, then the number of children in each family is :

1. A.3
2. B.4
3. C.5
4. D.6

A box 'A' contains 2 white, 3 red and 2 black balls. Another box 'B' contains 4 white, 2 red and 3 black balls. If two balls are drawn at random, without replacement, from a randomly selected box and one ball turns out to be white while the other ball turns out to be red, then the probability that both balls are drawn from box 'B' is :

1. A.$\frac{9}{16}$
2. B.$\frac{7}{16}$
3. C.$\frac{9}{32}$
4. D.$\frac{7}{8}$

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 :

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

From a group of 10 men and 5 women, four member committees are to be formed each of which must contain at least one woman. Then the probability for these committees to have more women than men, is :

1. A.$\frac{21}{220}$
2. B.$\frac{3}{11}$
3. C.$\frac{1}{11}$
4. D.$\frac{2}{23}$

Let E and F be two independent events. The probability that both E and F happen is $\frac{1}{12}$ and the probability that neither E nor F happens is $\frac{1}{2}$, then a value of $\frac{P(E)}{P(F)}$ is :

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

Three persons P, Q and R independently try to hit a target. I the probabilities of their hitting the target are $\frac{3}{4}$, $\frac{1}{2}$ and $\frac{5}{8}$ respectively, then the probability that the target is hit by P or Q but not by R is :

1. A.$\frac{21}{64}$
2. B.$\frac{9}{64}$
3. C.$\frac{15}{64}$
4. D.$\frac{39}{64}$

An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :

1. A.$\frac{255}{256}$
2. B.$\frac{127}{128}$
3. C.$\frac{63}{64}$
4. D.$\frac{1}{2}$

Let two fair six-faced dice $A$ and $B$ be thrown simultaneously. If $E_1$ is the event that die $A$ shows up four, $E_2$ is the event that die $B$ shows up two and $E_3$ is the event that the sum of numbers on both dice is odd, then which of the following statements is $\mathbf{NOT}$ true?

1. A.$E_1\text{ and }{E}_2\text{ are independent.}$
2. B.$E_2\text{ and }{E}_3\text{ are independent.}$
3. C.$E_1\text{ and }{E}_3\text{ are independent.}$
4. D.$E_1,E_2\text{ and }{E}_3\text{ are independent.}$

If 12 different balls are to be placed in 3 identical boxes, then the probability that one of the boxes contains exactly 3 balls is :

1. A.$220{\left(\frac{1}{3}\right)}^{12}$
2. B.$22{\left(\frac{1}{3}\right)}^{11}$
3. C.$\frac{55}{3}{\left(\frac{2}{3}\right)}^{11}$
4. D.$55{\left(\frac{2}{3}\right)}^{10}$

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

1. A.independent but not equally likely.
2. B.independent and equally likely.
3. C.mutually exclusive and independent.
4. D.equally likely but not independent.

There are two balls in an urn. Each ball can be either white or black. If a white ball is put into the urn and there after a ball is drawn at random from the urn, then the probability that it is white is

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

A number $n$ is randomly selected from the set $\{1,2,3,\dots ,1000\}$. The probability that $\frac{{\sum }_{i=1}^n{i}^2}{{\sum }_{i=1}^{n}i}$ is an integer is

1. A.$0.331$
2. B.$0.333$
3. C.$0.334$
4. D.$0.332$

There are two urns. Urn A has 3 distinct red balls and urn B has 9 distinct blue balls. From each urn two balls are taken out at random and then transferred to the other. The number of ways in which this can be done is

1. A.36
2. B.66
3. C.108
4. D.3

Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\dots 20\}$. Statement - 1: The probability that the chosen numbers when arranged in some order will form an AP is $\frac{1}{85}$. Statement - 2: If the four chosen numbers form an AP, then the set of all possible values of common difference is $\{\pm 1,\pm 2,\pm 3,\pm 4,\pm 5\}$.

1. A.Statement - 1 is true, Statement - 2 is true; Statement - 2 is not a correct explanation for Statement - 1.
2. B.Statement - 1 is true, Statement - 2 is false.
3. C.Statement - 1 is false, Statement -2 is true.
4. D.Statement - 1 is true, Statement - 2 is true; Statement - 2 is a correct explanation for Statement - 1.

An urn contains nine balls of which three are red, four are blue and two are green. Three balls are drawn at random without replacement from the urn. The probability that the three balls have different colours is :

1. A.$\frac{2}{7}$
2. B.$\frac{1}{21}$
3. C.$\frac{1}{23}$
4. D.$\frac{1}{3}$

One ticket is selected at random from 50 tickets numbered $00,01,02,\dots ,49$. Then the probability that the sum of the digits on the selected ticket is 8, given that the product of these digits is zero, equals :

1. A.$\frac{1}{7}$
2. B.$\frac{5}{14}$
3. C.$\frac{1}{50}$
4. D.$\frac{1}{14}$

In a binomial distribution $B\left(n,p=\frac{1}{4}\right)$, if the probability of at least one success is greater than or equal to $\frac{9}{10}$, then $n$ is greater than :

1. A.$\frac{1}{{\log }_{10}^4+{\log }_{10}^3}$
2. B.$\frac{9}{{\log }_{10}^4-{\log }_{10}^3}$
3. C.$\frac{4}{{\log }_{10}^4-{\log }_{10}^3}$
4. D.$\frac{1}{{\log }_{10}^4-{\log }_{10}^3}$

At a telephone enquiry system the number of phone calls regarding relevant enquiry follow Poisson distribution with an average of 5 phone calls during 10 minute time intervals. The probability that there is at the most one phone call during a 10-minute time period is :

1. A.$\frac{6}{5^e}$
2. B.$\frac{5}{6}$
3. C.$\frac{6}{55}$
4. D.$\frac{6}{e^5}$

Three houses are available in a locality. Three persons apply for the houses. Each applies for one house without consulting others. The probability that all the three apply for the same house is :

1. A.$\frac{2}{9}$
2. B.$\frac{1}{9}$
3. C.$\frac{8}{9}$
4. D.$\frac{7}{9}$

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

1. A.equally likely and mutually exclusive
2. B.equally likely but not independent
3. C.independent but not equally likely
4. D.mutually exclusive and independent

The probability that $A$ speaks truth is $\frac{4}{5}$, while the probability for $B$ is $\frac{3}{4}$. The probability that they contradict each other when asked to speak on a fact is :

1. A.$\frac{4}{5}$
2. B.$\frac{1}{5}$
3. C.$\frac{7}{20}$
4. D.$\frac{3}{20}$

A problem in mathematics is given to three students $A,B,C$ and their respective probability of solving the problem is $\frac{1}{2},\frac{1}{3}$ and $\frac{1}{4}$. Probability that the problem is solved is :

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

$A$ and $B$ are events such that $P(A\cup B)=3/4,P(A\cap B)=1/4,P(\overline{A})=2/3$ then $P(\overline{A}\cap B)$ is :

1. A.$5/12$
2. B.$3/8$
3. C.$5/8$
4. D.$1/4$