Probability · JEE Main previous year questions

Box I contains 30 cards numbered 1 to 30 and Box II contains 20 cards numbered 31 to 50. A box is selected at random and a card is drawn from it. The number on the card is found to be a non-prime number. The probability that the card was drawn from Box I is :

1. A.$\frac{8}{17}$
2. B.$\frac{2}{3}$
3. C.$\frac{2}{5}$
4. D.$\frac{4}{17}$

Let $E^C$ denote the complement of an event $E$. Let $E_1,E_2$ and $E_3$ be any pairwise independent events with $P(E_1)>0$ and $P(E_1\cap E_2\cap E_3)=0$. Then $P(E_2^C\cap E_3^C/E_1)$ is equal to :

1. A.$P(E_3^C)-P(E_2)$
2. B.$P(E_2^C)+P(E_3)$
3. C.$P(E_3^C)-P(E_2^C)$
4. D.$P(E_3)-P(E_2^C)$

In a box, there are 20 cards, out of which 10 are labelled as A and the remaining 10 are labelled as B. Cards are drawn at random, one after the other and with replacement, till a second A-card is obtained. The probability that the second A-card appears before the third B-card is :

1. A.$\frac{13}{16}$
2. B.$\frac{11}{16}$
3. C.$\frac{15}{16}$
4. D.$\frac{9}{16}$

A random variable X has the following probability distribution : $$\begin{array}{cccccc}\mathrm{X}:& 1& 2& 3& 4& 5\\ \mathrm{P}(\mathrm{X}):& {\mathrm{K}}^2& 2\mathrm{K}& \mathrm{K}& 2\mathrm{K}& 5{\mathrm{K}}^2\end{array}$$ Then $\mathrm{P}(\mathrm{X}>2)$ is equal to :

1. A.$\frac{1}{6}$
2. B.$\frac{7}{12}$
3. C.$\frac{1}{36}$
4. D.$\frac{23}{36}$

If 10 different balls are to be placed in 4 distinct boxes at random, then the probability that two of these boxes contain exactly 2 and 3 balls is :

1. A.$\frac{965}{2^{11}}$
2. B.$\frac{965}{2^{10}}$
3. C.$\frac{945}{2^{11}}$
4. D.$\frac{945}{2^{10}}$

Let A and B be two independent events such that $P(A)=\frac{1}{3}$ and $P(B)=\frac{1}{6}$. Then, which of the following is TRUE?

1. A.$P\left(\frac{A}{A\cup B}\right)=\frac{1}{4}$
2. B.$P\left(\frac{A}{B}\right)=\frac{2}{3}$
3. C.$P\left(\frac{A^{\prime }}{B^{\prime }}\right)=\frac{1}{3}$
4. D.$P\left(\frac{A}{B^{\prime }}\right)=\frac{1}{3}$

Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{1}{2}$ , then the probability of both of them occur together is :

1. A.0.20
2. B.0.02
3. C.0.01
4. D.0.10

An unbiased coin is tossed 5 times. Suppose that a variable X is assigned the value of k when k consecutive heads are obtained for $k=3,4,5$, otherwise X takes the value -1. Then the expected value of X, is :

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

In a workshop, there are five machines and the probability of any one of them to be out of service on a day is $\frac{1}{4}$. If the probability that at most two machines will be out of service on the same day is ${\left(\frac{3}{4}\right)}^{3}k$, then $k$ is equal to :

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

If three of the six vertices of a regular hexagon are chosen at random, then the probability that the triangle formed with these chosen vertices is equilateral is :

1. A.$\frac{1}{10}$
2. B.$\frac{3}{10}$
3. C.$\frac{3}{20}$
4. D.$\frac{1}{5}$

A person throws two fair dice. He wins Rs. 15 for throwing a doublet (same numbers on the two dice), wins Rs. 12 when the throw results in the sum of 9, and loses Rs. 6 for any other outcome on the throw. Then the expected gain/loss (in Rs.) of the person is :

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

For an initial screening of an admission test, a candidate is given fifty problems to solve. If the probability that the candidate solve any problem is $\frac{4}{5}$, then the probability that he is unable to solve less than two problems is :

1. A.$\frac{164}{25}{\left(\frac{1}{5}\right)}^{48}$
2. B.$\frac{316}{25}{\left(\frac{4}{5}\right)}^{48}$
3. C.$\frac{201}{5}{\left(\frac{1}{5}\right)}^{49}$
4. D.$\frac{54}{5}{\left(\frac{4}{5}\right)}^{49}$

Assume that each born child is equally likely to be a boy or a girl. If two families have two children each, then the conditional probability that all children are girls given that at least two are girls is :

1. A.$\frac{1}{10}$
2. B.$\frac{1}{17}$
3. C.$\frac{1}{11}$
4. D.$\frac{1}{12}$

Minimum number of times a fair coin must be tossed so that the probability of getting at least one head is more than 99% is :

1. A.6
2. B.5
3. C.8
4. D.7

Four persons can hit a target correctly with probabilities $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and $\frac{1}{8}$ respectively. If all hit at the target independently, then the probability that the target would be hit, is :

1. A.$\frac{25}{32}$
2. B.$\frac{25}{192}$
3. C.$\frac{1}{192}$
4. D.$\frac{7}{32}$

The minimum number of times one has to toss a fair coin so that the probability of observing at least one head is at least 90% is :

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

In a random experiment, a fair die is rolled until two fours are obtained in succession. The probability that the experiment will end in the fifth throw of the die is equal to :

1. A.$\frac{200}{6^5}$
2. B.$\frac{225}{6^5}$
3. C.$\frac{150}{6^5}$
4. D.$\frac{175}{6^5}$

There are m men and two women participating in a chess tournament. Each participant plays two games with every other participant. If the number of games played by the men between themselves exceeds the number of games played between the men and the women by 84, then the value of m is :

1. A.12
2. B.9
3. C.7
4. D.11

In a game, a man wins Rs. 100 if he gets 5 or 6 on a throw of a fair die and loses Rs. 50 for getting any other number on the die. If he decides to throw the die either till he gets a five or a six or to a maximum of three throws, then his expected gain/loss (in rupees) is :

1. A.$\frac{400}{3}$ loss
2. B.0
3. C.$\frac{400}{9}$ loss
4. D.$\frac{400}{3}$ gain

In a class of 60 students, 40 opted for NCC, 30 opted for NSS and 20 opted for both NCC and NSS. If one of these students is selected at random, then the probability that the students selected has opted neither for NCC nor for NSS is :

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

Let $S=\{1,2,\dots ,20\}$. A subset B of S is said to be "nice", if the sum of the elements of B is 203. Then the probability that a randomly chosen subset of S is "nice" is :

1. A.$\frac{5}{2^{20}}$
2. B.$\frac{7}{2^{20}}$
3. C.$\frac{4}{2^{20}}$
4. D.$\frac{6}{2^{20}}$

A bag contains 30 white balls and 10 red balls. 16 balls are drawn one by one randomly from the bag with replacement. If X be the number of white balls drawn, then $\left(\frac{\text{mean of }X}{\text{standard deviation of }X}\right)$ is equal to :

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

The value of ${\int }_0^{\pi }|\cos x{|}^{3}dx$ is :

1. A.$\frac{4}{3}$
2. B.$-\frac{4}{3}$
3. C.0
4. D.$\frac{2}{3}$

Two cards are drawn successively with replacement from a well-shuffled deck of 52 cards. Let X denote the random variable of number of aces obtained in the two drawn cards. Then $P(X=1)+P(X=2)$ equals :

1. A.$\frac{25}{169}$
2. B.$\frac{49}{169}$
3. C.$\frac{24}{169}$
4. D.$\frac{52}{169}$

Let A, B and C be three events, which are pair-wise independent and $\overline{E}$ denotes the complement of an event E. If $P(A\cap B\cap C)=0$ and $P(C)>0$, then $P\left[\left(\overline{A}\cap \overline{B}\right)|C\right]$ is equal to :

1. A.$P\left(\overline{A}\right)-P(B)$
2. B.$P(A)+P\left(\overline{B}\right)$
3. C.$P\left(\overline{A}\right)-P\left(\overline{B}\right)$
4. D.$P\left(\overline{A}\right)+P\left(\overline{B}\right)$