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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 51-100 of 372

  1. If a random variable $X$ follows the Binomial distribution $B(33,p)$ such that $3P(X=0)=P(X=1)$, then the value of $\frac{P(X=15)}{P(X=18)}-\frac{P(X=16)}{P(X=17)}$ is …
  2. The probability that a randomly selected 2-digit number belongs to the set $\{n\in\mathbf{N}:(2^n-2)\text{ is a multiple of }3\}$ is equal to :
  3. Given three identical bags each containing $10$ balls, whose colours are as follows : Bag I : Red $3$, Blue $2$, Green $5$ Bag II : Red $4$, Blue $3$, Green $3$ Bag III …
  4. If the sum and the product of mean and variance of a binomial distribution are $24$ and $128$ respectively, then the probability of one or two successes is :
  5. Bag $A$ contains $2$ white, $1$ black and $3$ red balls and bag $B$ contains $3$ black, $2$ red and $n$ white balls. One bag is chosen at random and $2$ balls drawn from …
  6. Let $X$ be a random variable having binomial distribution $B(7,p)$. If $P(X=3)=5P(X=4)$, then the sum of the mean and the variance of $X$ is :
  7. A student appeared in an examination consisting of $8$ true-false type questions. The student guesses the answers with equal probability. The smallest value of $n$, so …
  8. Let $A$ and $B$ be two events such that $P(B|A)=\frac25$, $P(A|B)=\frac17$ and $P(A\cap B)=\frac19$. Consider (S1) $P(A'\cup B)=\frac56$, (S2) $P(A'\cap B')=\frac1{18}$. …
  9. Let $X$ have a binomial distribution $B(n,p)$ such that the sum and the product of the mean and variance of $X$ are $24$ and $128$ respectively. If …
  10. If $A$ and $B$ are two events such that $P(A)=0.7$, $P(B)=0.4$ and $P(A\cap\bar{B})=0.5$, where $\bar{B}$ denotes the complement of $B$, then …
  11. The probabilities of three events $A$, $B$ and $C$ are given by $P(A)=0.6$, $P(B)=0.4$ and $P(C)=0.5$. If $P(A\cup B)=0.8$, $P(A\cap C)=0.3$, $P(A\cap B\cap C)=0.2$ …
  12. 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 …
  13. A random variable $X$ has the following probability distribution : $X$: $1,2,3,4,5$ and $P(X)$: $K^2,2K,K,2K,5K^2$. Then $P(X>2)$ is equal to :
  14. Let there be three independent events $E_1$, $E_2$ and $E_3$. The probability that only $E_1$ occurs is $\alpha$, only $E_2$ occurs is $\beta$ and only $E_3$ occurs is …
  15. 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 …
  16. Let $B_i$ ($i=1,2,3$) be three independent events in a sample space. The probability that only $B_1$ occur is $\alpha$, only $B_2$ occurs is $\beta$ and only $B_3$ …
  17. Let $S=\{1,2,3,\ldots,2022\}$. Then the probability, that a randomly chosen number $n$ from the set $S$ such that $\mathrm{HCF}(n,2022)=1$, is :
  18. If an unbiased dice is rolled thrice, then the probability of getting a greater number in the $i^{\text{th}}$ roll than the number obtained in the $(i-1)^{\text{th}}$ …
  19. The mean and variance of a binomial distribution are $\alpha$ and $\frac\alpha3$ respectively. If $P(X=1)=\frac{4}{243}$, then $P(X=4\text{ or }5)$ is equal to :
  20. The probability that a randomly chosen one-one function from the set $\{a,b,c,d\}$ to the set $\{1,2,3,4,5\}$ satisfies $f(a)+2f(b)-f(c)=f(d)$ is :
  21. A biased die is marked with numbers $2,4,8,16,32,32$ on its faces and the probability of getting a face with mark $n$ is $\frac{1}{n}$. If the die is thrown thrice, then …
  22. The probability, of forming a $12$ persons committee from $4$ engineers, $2$ doctors and $10$ professors containing at least $3$ engineers and at least $1$ doctor, is
  23. 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 …
  24. Let $E_1$ and $E_2$ be two events such that the conditional probabilities $P(E_1|E_2)=\frac{1}{2}$, $P(E_2|E_1)=\frac{3}{4}$ and $P(E_1\cap E_2)=\frac{1}{8}$. Then :
  25. If the numbers appeared on the two throws of a fair six faced die are $\alpha$ and $\beta$, then the probability that $x^2+\alpha x+\beta>0$, for all $x\in\mathbf{R}$ …
  26. There are three bags $X$, $Y$ and $Z$. Bag $X$ contains $5$ one-rupee coins and $4$ five-rupee coins; Bag $Y$ contains $4$ one-rupee coins and $5$ five-rupee coins and …
  27. A die is thrown two times and the sum of the scores appearing on the die is observed to be a multiple of $4$. Then the conditional probability that the score $4$ has …
  28. Let a biased coin be tossed $5$ times. If the probability of getting $4$ heads is equal to the probability of getting $5$ heads, then the probability of getting atmost …
  29. From a month of $31$ days, $3$ different dates are selected at random. If the probability that these dates are in an increasing A.P. is equal to $\frac{a}{b}$, where …
  30. Let $S$ be the sample space of all five digit numbers. It $p$ is the probability that a randomly selected number from $S$, is a multiple of $7$ but not divisible by $5$ …
  31. Let $A$ and $B$ be two events such that the probability that exactly one of them occurs is $\frac25$ and the probability that $A$ or $B$ occurs is $\frac12$, then the …
  32. If the probability that the random variable $X$ takes the value $x$ is given by $P(X=x)=k(x+1)3^{-x}$, $x=0,1,2,3,\ldots$, where $k$ is a constant, then $P(X\ge3)$ is …
  33. Let $A$ and $B$ be two independent events such that $P(A)=\frac13$ and $P(B)=\frac16$. Then, which of the following is TRUE ?
  34. Let $X$ be a binomially distributed random variable with mean $4$ and variance $\frac43$. Then, $54P(X\le2)$ is equal to
  35. A random variable $X$ has the following probability distribution: $X$ : $0,1,2,3,4$ $P(X)$ : $k,2k,4k,6k,8k$ The value of $P(1<X<4\mid X\le2)$ is equal to :
  36. A fair coin is tossed $n$-times such that the probability of getting at least one head is at least $0.9$. Then the minimum value of $n$ is $\_\_\_\_\_\_$ .
  37. Five numbers $x_1,x_2,x_3,x_4,x_5$ are randomly selected from the numbers $1,2,3,\ldots,18$ and are arranged in the increasing order ($x_1<x_2<x_3<x_4<x_5$). The …
  38. The probability that a randomly chosen $5$-digit number is made from exactly two digits is :
  39. Let $E_1,E_2,E_3$ be three mutually exclusive events such that $P(E_1)=\frac{2+3p}{6}$, $P(E_2)=\frac{2-p}{8}$ and $P(E_3)=\frac{1-p}{2}$. If the maximum and minimum …
  40. In a game two players $A$ and $B$ take turns in throwing a pair of fair dice starting with player $A$ and total of scores on the two dice, in each throw is noted. $A$ …
  41. A six faced die is biased such that $3\times P(\text{a prime number})=6\times P(\text{a composite number})=2\times P(1)$. Let $X$ be a random variable that counts the …
  42. If a point $A(x,y)$ lies in the region bounded by the $y$-axis, straight lines $2y+x=6$ and $5x-6y=30$, then the probability that $y<1$ is
  43. If $A$ and $B$ are two events such that $P(A)=\frac13$, $P(B)=\frac15$ and $P(A\cup B)=\frac12$, then $P(A|B')+P(B|A')$ is equal to
  44. A bag contains $19$ unbiased coins and one coin with head on both sides. One coin drawn at random is tossed and head turns up. If the probability that the drawn coin was …
  45. If a random variable $X$ follows the Binomial distribution $B(5,p)$ such that $P(X=0)=P(X=1)$, then $\frac{P(X=2)}{P(X=3)}$ is equal to :
  46. 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 …
  47. Three balls are drawn at random from a bag containing $5$ blue and $4$ yellow balls. Let the random variables $X$ and $Y$ respectively denote the number of blue and …
  48. A fair $n$ ($n>1$) faces die is rolled repeatedly until a number less than $n$ appears. If the mean of the number of tosses required is $\frac{n}{9}$, then $n$ is equal …
  49. The probability of a man hitting a target is $\frac1{10}$. The least number of shots required, so that the probability of his hitting the target at least once is greater …
  50. An electric instrument consists of two units. Each unit must function independently for the instrument to operate. The probability that the first unit functions is $0.9$ …

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