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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 101-150 of 372

  1. The probability distribution of random variable $X$ is given by : $X$ : $1,2,3,4,5$ $P(X)$ : $K,2K,2K,3K,K$ Let $p=P(1<X<4\mid X<3)$. If $5p=\lambda K$, then $\lambda$ …
  2. In an examination, there are $10$ true-false type questions. Out of $10$, a student can guess the answer of $4$ questions correctly with probability $\frac{3}{4}$ and …
  3. Let $S=\{E_1,E_2,\ldots,E_8\}$ be a sample space of a random experiment such that $P(E_n)=\frac{n}{36}$ for every $n=1,2,\ldots,8$. Then the number of elements in the …
  4. If the probability that a randomly chosen $6$-digit number formed by using digits $1$ and $8$ only is a multiple of $21$ is $p$, then $96p$ is equal to $\_\_\_\_\_\_$ .
  5. 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 $ax^{2}+bx+c=0$ …
  6. The sum and product of the mean and variance of a binomial distribution are $82.5$ and $1350$ respectively. Then the number of trials in the binomial distribution is …
  7. All five letter words are made using all the letters $A,B,C,D,E$ and arranged as in an English dictionary with serial numbers. Let the word at serial number $n$ be …
  8. 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 …
  9. 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 …
  10. Let the probability of getting head for a biased coin be $\frac{1}{4}$. It is tossed repeatedly until a head appears. Let $N$ be the number of tosses required. If the …
  11. A fair die is tossed repeatedly until a six is obtained. Let $X$ denote the number of tosses required and let $a=P(X=3)$, $b=P(X\ge3)$ and $c=P(X\ge6|X>3)$. Then …
  12. Three distinct numbers are selected randomly from the set $\{1,2,3,\ldots,40\}$. If the probability, that the selected numbers are in an increasing G.P., is …
  13. Four fair dice are thrown independently $27$ times. Then the expected number of times, at least two dice show up a three or a five, is ________.
  14. The probability distribution of a random variable $X$ is $P(X=i)=\frac1{2^i}$, $i=1,2,3,\ldots$. The variance of $X$ is equal to ________.
  15. A card from a pack of $52$ cards is lost. From the remaining $51$ cards, $n$ cards are drawn and are found to be spades. If the probability of the lost card to be a …
  16. Let $A$ and $B$ two events such that $P\left(\overline{A\cup B}\right)=\frac16$, $P(A\cap B)=\frac14$ and $P\left(\bar A\right)=\frac14$, where $\bar A$ stands for …
  17. Five horses are in a race. Mr. $A$ selects two of the horses at random and bets on them. The probability that Mr. $A$ selected the winning horse is :
  18. An experiment succeeds twice as often as it fails. The probability of at least $5$ successes in the six trials of this experiment is
  19. One ticket is selected at random from $50$ tickets numbered $00,01,02,\ldots,49$. Then the probability that the sum of the digits on the selected ticket is $8$, given …
  20. If $A$ and $B$ are any two events such that $P(A)=\frac25$ and $P(A\cap B)=\frac{3}{20}$, then the conditional probability $P(A\mid(A'\cup B'))$, where $A'$ denotes the …
  21. If the events $A$ and $B$ are mutually exclusive events such that $P(A)=\frac{3x+1}{3}$ and $P(B)=\frac{1-x}{4}$, then the set of possible values of $x$ lies in the …
  22. 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 …
  23. A random variable $X$ has Poisson distribution with mean $2$. Then $P(X>1.5)$ equals :
  24. $A,B,C$ try to hit a target simultaneously but independently. Their respective probabilities of hitting the targets are $\frac34,\frac12,\frac58$. The probability that …
  25. Given two independent events, if the probability that exactly one of them occurs is $\frac{26}{49}$ and the probability that none of them occurs is $\frac{15}{49}$, then …
  26. A problem in mathematics is given to three students $A,B,C$ and their respective probability of solving the problem is $\frac12$, $\frac13$ and $\frac14$. Probability …
  27. 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 …
  28. Three persons $P$, $Q$ and $R$ independently try to hit a target. If the probabilities of their hitting the target are $\frac34$, $\frac12$ and $\frac58$ respectively …
  29. Let $S=\{1,2,\ldots,20\}$. 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 …
  30. Let $A$ and $E$ be any two events with positive probabilities. Statement I: $P(E/A)\ge P(A/E)P(E)$. Statement II: $P(A/E)\ge P(A\cap E)$.
  31. 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 …
  32. Let $X$ be a set containing $10$ elements and $P(X)$ be its power set. If $A$ and $B$ are picked up at random from $P(X)$, with replacement, then the probability that …
  33. An urn contains $5$ red and $2$ green balls. A ball is drawn at random from the urn. If the drawn ball is green, then a red ball is added to the urn and if the drawn …
  34. If $C$ and $D$ are two events such that $C\subset D$ and $P(D)\neq0$, then the correct statement among the following is :
  35. The probability that $A$ speaks truth is $\frac45$, while the probability for $B$ is $\frac34$. The probability that they contradict each other when asked to speak on a …
  36. Four numbers are chosen at random (without replacement) from the set $\{1,2,3,\ldots,20\}$. Statement - 1: The probability that the chosen numbers when arranged in some …
  37. 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 $P(C)>0$ …
  38. A bag contains $4$ red and $6$ black balls. A ball is drawn at random from the bag, its color is observed and this ball along with two additional balls of the same color …
  39. The probability of a man hitting a target is $\frac25$. He fires at the target $k$ times ($k$, a given number). Then the minimum $k$, so that the probability of hitting …
  40. A die is thrown. Let $A$ be the event that the number obtained is greater than $3$. Let $B$ be the event that the number obtained is less than $5$. Then $P(A\cup B)$ is :
  41. Let $E$ and $F$ be two independent events. The probability that $E$ and $F$ happen is $\frac1{12}$ and the probability that neither $E$ nor $F$ happens is $\frac12$ …
  42. Let $A$ and $B$ be two events such that $P\left(\overline{A\cup B}\right)=\frac16$, $P(A\cap B)=\frac14$ and $P\left(\bar A\right)=\frac14$, where $\bar A$ stands for …
  43. $A$ and $B$ are events such that $P(A\cup B)=3/4$, $P(A\cap B)=1/4$, $P(\bar A)=2/3$ then $P(\bar A\cap B)$ is :
  44. 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 :
  45. A player $X$ has a biased coin whose probability of showing heads is $p$ and a player $Y$ has a fair coin. They start playing a game with their own coins and play …
  46. 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 …
  47. An unbiased coin is tossed eight times. The probability of obtaining at least one head and at least one tail is :
  48. If two different numbers are taken from the set $\{0,1,2,3,\ldots,10\}$, then the probability that their sum as well as absolute difference are both multiple of $4$, is :
  49. 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 …
  50. A number $x$ is chosen at random from the set $\{1,2,3,4,\ldots,100\}$. Define the event $A$ = the chosen number $x$ satisfies $\frac{(x-10)(x-50)}{(x-30)}\ge0$. Then …

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