Probability · JEE Main previous year questions

Let a die be rolled $n$ times. Let the probability of getting odd numbers seven times be equal to the probability of getting odd numbers nine times. If the probability of getting even numbers twice is $\frac{k}{2^{15}}$, then k is equal to :

1. A.15
2. B.60
3. C.30
4. D.90

In a bolt factory, machines $A,B$ and $C$ manufacture respectively $20\%,30\%$ and $50\%$ of the total bolts. Of their output $3,4$ and $2$ percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found the defective, then the probability that it is manufactured by the machine $C$ is :

1. A.$\frac{2}{7}$
2. B.$\frac{9}{28}$
3. C.$\frac{5}{14}$
4. D.$\frac{3}{7}$

If the probability that the random variable $X$ takes values $x$ is given by $\mathrm{P}(X=x)=\mathrm{k}(x+1)3^{-x},x=0,1,2,3,\dots$, where $\mathrm{k}$ is a constant, then $\mathrm{P}(X\ge 2)$ is equal to :

1. A.$\frac{7}{18}$
2. B.$\frac{20}{27}$
3. C.$\frac{7}{27}$
4. D.$\frac{11}{18}$

A pair of dice is thrown 5 times. For each throw, a total of 5 is considered a success. If the probability of at least 4 successes is $\frac{k}{3^{11}}$, then $k$ is equal to :

1. A.82
2. B.164
3. C.123
4. D.75

Three dice are rolled. If the probability of getting different numbers on the three dice is $\frac{p}{q}$, where $p$ and $q$ are co-prime, then $q-p$ is equal to :

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

A bag contains 6 balls. Two balls are drawn from it at random and both are found to be black. The probability that the bag contains at least 5 black balls is :

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

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 $\mathrm{P}(\mathrm{A})=\frac{11}{36}$, then a is equal to ___________.

If an unbiased die, marked with $-2,-1,0,1,2,3$ on its faces, is thrown five times, then the probability that the product of the outcomes is positive, is :

1. A.$\frac{27}{288}$
2. B.$\frac{521}{2592}$
3. C.$\frac{440}{2592}$
4. D.$\frac{881}{2592}$

Fifteen football players of a club-team are given 15 T-shirts with their names written on the backside. If the players pick up the T-shirts randomly, then the probability that at least 3 players pick the correct T-shirt is :

1. A.$\frac{1}{6}$
2. B.$\frac{2}{15}$
3. C.$\frac{5}{24}$
4. D.0.08

Let $\mathrm{S}=\{w_1,w_2,\dots \}$ be the sample space associated to a random experiment. Let $P(w_n)=\frac{P(w_{n-1})}{2},n\ge 2$. Let $A=\{2k+3l:k,l\in N\}$ and $B=\{w_n:n\in A\}$. Then P(B) is equal to :

1. A.$\frac{3}{32}$
2. B.$\frac{1}{32}$
3. C.$\frac{1}{16}$
4. D.$\frac{3}{64}$

Let N be the sum of the numbers appeared when two fair dice are rolled and let the probability that $N-2,\sqrt{3N},N+2$ are in geometric progression be $\frac{k}{48}$. Then the value of k is :

1. A.8
2. B.16
3. C.2
4. D.4

25% of the population are smokers. A smoker has 27 times more chances to develop lung cancer than a non smoker. A person is diagnosed with lung cancer and the probability that this person is a smoker is $\frac{k}{10}$. Then the value of k is ____.

Let N denote the number that turns up when a fair die is rolled. If the probability that the system of equations $x+y+z=1$ $2x+Ny+2z=2$ $3x+3y+Nz=3$ has unique solution is $\frac{k}{6}$, then the sum of value of k and all possible values of N is :

1. A.18
2. B.21
3. C.20
4. D.19

Three urns A, B and C contain 4 red, 6 black; 5 red, 5 black; and $\lambda$ red, 4 black balls respectively. One of the urns is selected at random and a ball is drawn. If the ball drawn is red and the probability that it is drawn from urn C is 0.4 then the square of the length of the side of the largest equilateral triangle, inscribed in the parabola $y^2=\lambda x$ with one vertex at the vertex of the parabola, is :

Let $S=\{1,2,3,\dots ,2022\}$. Then the probability, that a randomly chosen number n from the set S such that $\mathrm{HCF}(n,2022)=1$, is :

1. A.$\frac{128}{1011}$
2. B.$\frac{166}{1011}$
3. C.$\frac{127}{337}$
4. D.$\frac{112}{337}$

Bag I contains 3 red, 4 black and 3 white balls and Bag II contains 2 red, 5 black and 2 white balls. One ball is transferred from Bag I to Bag II and then a ball is drawn from Bag II. The ball so drawn is found to be black in colour. Then the probability, that the transferred ball is red, is :

1. A.$\frac{4}{9}$
2. B.$\frac{5}{18}$
3. C.$\frac{1}{6}$
4. D.$\frac{3}{10}$

Out of 60% female and 40% male candidates appearing in an exam, 60% candidates qualify it. The number of females qualifying the exam is twice the number of males qualifying it. A candidate is randomly chosen from the qualified candidates. The probability, that the chosen candidate is a female, is :

1. A.$\frac{2}{3}$
2. B.$\frac{11}{16}$
3. C.$\frac{23}{32}$
4. D.$\frac{13}{16}$

Let A and B be two events such that $P(B\mid A)=\frac{2}{5}$, $P(A\mid B)=\frac{1}{7}$ and $P(A\cap B)=\frac{1}{9}$. Consider (S1) $P(A^{\prime }\cup B)=\frac{5}{6}$, (S2) $P(A^{\prime }\cap B^{\prime })=\frac{1}{18}$ Then :

1. A.Both (S1) and (S2) are true
2. B.Both (S1) and (S2) are false
3. C.Only (S1) is true
4. D.Only (S2) is true

A bag contains 4 white and 6 black balls. Three balls are drawn at random from the bag. Let X be the number of white balls, among the drawn balls. If ${\sigma }^2$ is the variance of X, then $100{\sigma }^2$ is equal to _________.

Let $S$ be the sample space of all five digit numbers. If $p$ is the probability that a randomly selected number from $S$, is a multiple of 7 but not divisible by 5 , then $9p$ is equal to :

1. A.1.0146
2. B.1.2085
3. C.1.0285
4. D.1.1521

A six faced die is biased such that $3\times \mathrm{P}(\text{a prime number})=6\times \mathrm{P}(\text{a composite number})=2\times \mathrm{P}(1)$. Let $\mathrm{X}$ be a random variable that counts the number of times one gets a perfect square on some throws of this die. If the die is thrown twice, then the mean of $\mathrm{X}$ is :

1. A.$\frac{3}{11}$
2. B.$\frac{5}{11}$
3. C.$\frac{7}{11}$
4. D.$\frac{8}{11}$

If $A$ and $B$ are two events such that $P(A)=\frac{1}{3}$, $P(B)=\frac{1}{5}$ and $P(A\cup B)=\frac{1}{2}$, then $P(A\mid B^{\prime })+P(B\mid A^{\prime })$ is equal to :

1. A.$\frac{3}{4}$
2. B.$\frac{5}{8}$
3. C.$\frac{5}{4}$
4. D.$\frac{7}{8}$

Let $f(t)={\int }_0^t{\mathrm{e}}^{x^3}\left(\frac{x^8}{(x^6+2x^3+2{)}^2}\right)\mathrm{d}x$. If $f(1)+f^{\prime }(1)=\alpha \mathrm{e}-\frac{1}{6}$, then the value of $150\alpha$ is equal to ____ .

The probability distribution of X is : $$\begin{array}{ccccc}\mathrm{X}& 0& 1& 2& 3\\ \mathrm{P}(\mathrm{X})& \frac{1-d}{4}& \frac{1+2d}{4}& \frac{1-4d}{4}& \frac{1+3d}{4}\end{array}$$ For the minimum possible value of d, sixty times the mean of X is equal to ____ .

Five numbers $x_1,x_2,x_3,x_4,x_5$ are randomly selected from the numbers $1,2,3,.......,18$ and are arranged in the increasing order $(x_1<x_2<x_3<x_4<x_5)$. The probability that $x_2=7$ and $x_4=11$ is :

1. A.$\frac{1}{136}$
2. B.$\frac{1}{72}$
3. C.$\frac{1}{68}$
4. D.$\frac{1}{34}$