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Statistics & Probability

Probability: JEE Main & Advanced PYQs

372 previous year questions on Probability from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 1-50 of 372

  1. Four dice are thrown simultaneously and the numbers shown on these dice are recorded in $2\times2$ matrices. The probability that such formed matrices have all different …
  2. A dice is thrown twice. If $A$ is the event that the sum of numbers appearing on them is $7$ and $B$ is the event that number $5$ appears at least once, then $P(B|A)$ is …
  3. Let in a Binomial distribution, consisting of $5$ independent trials, probabilities of exactly $1$ and $2$ successes be $0.4096$ and $0.2048$ respectively. Then the …
  4. Two players $A$ and $B$ play a game. Player $A$ first throws a fair die and if $5$ or $6$ appears, he wins; otherwise player $B$ throws a fair coin and wins if head …
  5. When a missile is fired from a ship, the probability that it is intercepted is $\frac{1}{3}$ and the probability that the missile hits the target, given that it is not …
  6. Four persons work independently of each other on a task. If the probabilities that they will complete the task are : $\frac{1}{2},\frac{1}{3},\frac{1}{4}$ and …
  7. Let $A$ denote the event that a $6$-digit integer formed by $0,1,2,3,4,5,6$ without repetitions, be divisible by $3$. Then probability of event $A$ is equal to :
  8. Let $9$ distinct balls be distributed among $4$ boxes, $B_1,B_2,B_3$ and $B_4$. If the probability that $B_3$ contains exactly $3$ balls is …
  9. A company has two plants $A$ and $B$ to manufacture motorcycles. $60\%$ motorcycles are manufactured at plant $A$ and the remaining are manufactured at plant $B$. $80\%$ …
  10. 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 …
  11. A bag contains $6$ white and $4$ black balls. A die is rolled once and the number of balls equal to the number obtained on the die are drawn from the bag at random. The …
  12. A fair die is tossed until six is obtained on it. Let $X$ be the number of required tosses, then the conditional probability $P(X\ge5\mid X>2)$ is :
  13. A coin is biased so that the head is $3$ times as likely to occur as tail. This coin is tossed until a head or three tails occur. If $X$ denotes the number of tosses of …
  14. Let $S=\{M=[a_{ij}],\ a_{ij}\in\{0,1,2\},\ 1\le i,j\le2\}$ be a sample space and $A=\{M\in S:M\text{ is invertible}\}$ be an event. Then $P(A)$ is equal to
  15. A bag contains $6$ blue and $6$ green balls. Pairs of balls are drawn without replacement until the bag is empty. The probability that each drawn pair consists of one …
  16. Two fair dice are thrown. The numbers on them are taken as $\lambda$ and $\mu$, and a system of linear equations $x+y+z=5$ $x+2y+3z=\mu$ $x+3y+\lambda z=1$ is …
  17. Let $A=\{0,1,2,3,4,5\}$. Then the probability that a randomly selected onto function $f$ from $A$ to $A$ satisfies $f(1)+f(2)=f(3)$ is
  18. Each of the persons $A$ and $B$ independently tosses three fair coins. The probability that both of them get the same number of heads is :
  19. Let $X$ be a random variable such that the probability function of a distribution is given by $P(X=0)=\frac{1}{2}$, $P(X=j)=\frac{1}{3^{j}}$ $(j=1,2,3,\ldots,\infty)$. …
  20. A fair die is thrown until $2$ appears. Then the probability, that $2$ appears in even number of throws, is
  21. Two integers $x$ and $y$ are chosen with replacement from the set $\{0,1,2,3,\ldots,10\}$. Then the probability that $|x-y|>5$, is :
  22. A man throws a fair coin repeatedly. He gets $10$ points for each head he throws and $5$ points for each tail he throws. If the probability that he gets exactly $30$ …
  23. When a certain biased die is rolled, a particular face occurs with probability $\frac{1}{6}-x$ and its opposite face occurs with probability $\frac{1}{6}+x$. All other …
  24. 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 $k$ …
  25. Two dices are rolled. If both dices have six faces numbered $1,2,3,5,7$ and $11$, then the probability that the sum of the numbers on the top faces is less than or equal …
  26. The random variable $X$ follows binomial distribution $B(n,p)$, for which the difference of the mean and the variance is $1$. If $2P(X=2)=3P(X=1)$, then $n^{2}P(X>1)$ is …
  27. Let $A$ and $B$ be independent events such that $P(A)=p$, $P(B)=2p$. The largest value of $p$, for which $P(\text{exactly one of }A,B\text{ occurs})=\frac{5}{9}$, is :
  28. 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 …
  29. 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 …
  30. In a group of $400$ people, $160$ are smokers and non-vegetarian; $100$ are smokers and vegetarian and the remaining $140$ are non-smokers and vegetarian. Their chances …
  31. Let $A$ be a set of all $4$-digit natural numbers whose exactly one digit is $7$. Then the probability that a randomly chosen element of $A$ leaves remainder $2$ when …
  32. The probability that two randomly selected subsets of the set $\{1,2,3,4,5\}$ have exactly two elements in their intersection, is :
  33. 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 $4m-3n$ …
  34. If the probability that the random variable $X$ takes values $x$ is given by $P(X=x)=k(x+1)3^{-x}$, $x=0,1,2,3,\ldots$, where $k$ is a constant, then $P(X\ge2)$ is equal …
  35. 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
  36. An ordinary dice is rolled for a certain number of times. If the probability of getting an odd number $2$ times is equal to the probability of getting an even number $3$ …
  37. 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 :
  38. A seven digit number is formed using digits $3,3,4,4,4,5,5$. The probability, that number so formed is divisible by $2$, is :
  39. The probabilities that players $A$ and $B$ of a team are selected for the captaincy for a tournament are $0.6$ and $0.4$, respectively. If $A$ is selected the captain …
  40. 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 …
  41. 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
  42. The coefficients $a,b$ and $c$ of the quadratic equation, $ax^{2}+bx+c=0$ are obtained by throwing a dice three times. The probability that this equation has equal roots …
  43. 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 …
  44. Let $S=\{1,2,3,4,5,6\}$. Then the probability that a randomly chosen onto function $g$ from $S$ to $S$ satisfies $g(3)=2g(1)$ is :
  45. 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 …
  46. A fair coin is tossed a fixed number of times. If the probability of getting $7$ heads is equal to probability of getting $9$ heads, then the probability of getting $2$ …
  47. Bag $A$ contains $3$ white, $7$ red balls and Bag $B$ contains $3$ white, $2$ red balls. One bag is selected at random and a ball is drawn from it. The probability of …
  48. A candidate has to go to the examination centre to appear in an examination. The candidate uses only one means of transportation for the entire distance out of bus …
  49. The probability that a randomly chosen $2\times2$ matrix with all the entries from the set of first $10$ primes, is singular, is equal to :
  50. The probability, that in a randomly selected $3$-digit number at least two digits are odd, is

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