Probability · JEE Main previous year questions

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2022 17 questions

Q142022 · 29 Jul 2022 · Shift 1
Let $S = {1, 2, 3, \dots, 2022}$ . Then the probability, that a randomly chosen number n from the set S such that $HCF (n, 2022) = 1$ , is :
1. A. $\frac{128}{1011}$
2. B. $\frac{166}{1011}$
3. C. $\frac{127}{337}$
4. D. $\frac{112}{337}$
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Answer: D
Q112022 · 29 Jul 2022 · Shift 2
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. $\frac{3}{10}$
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Answer: B
Q82022 · 28 Jul 2022 · Shift 1
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. $\frac{13}{16}$
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Answer: A
Q142022 · 28 Jul 2022 · Shift 2
Let A and B be two events such that $P (B ∣ A) = \frac{2}{5}$ , $P (A ∣ B) = \frac{1}{7}$ and $P (A \cap B) = \frac{1}{9}$ . Consider (S1) $P (A^{'} \cup B) = \frac{5}{6}$ , (S2) $P (A^{'} \cap B^{'}) = \frac{1}{18}$ 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
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Answer: A
Q242022 · 28 Jul 2022 · Shift 2· Numerical answer
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 $σ^{2}$ is the variance of X, then $100 σ^{2}$ is equal to _________.
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Answer: 56
Q122022 · 27 Jul 2022 · Shift 1
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 $9 p$ is equal to :
1. A.1.0146
2. B.1.2085
3. C.1.0285
4. D.1.1521
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Answer: C
Q122022 · 27 Jul 2022 · Shift 2
A six faced die is biased such that $3 \times P (a prime number) = 6 \times P (a composite number) = 2 \times P (1)$ . Let $X$ 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 $X$ is :
1. A. $\frac{3}{11}$
2. B. $\frac{5}{11}$
3. C. $\frac{7}{11}$
4. D. $\frac{8}{11}$
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Answer: D
Q62022 · 25 Jul 2022 · Shift 2
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 $P (A ∣ B^{'}) + P (B ∣ A^{'})$ is equal to :
1. A. $\frac{3}{4}$
2. B. $\frac{5}{8}$
3. C. $\frac{5}{4}$
4. D. $\frac{7}{8}$
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Answer: B
Q242022 · 30 Jun 2022 · Shift 1· Numerical answer
Let $f (t) = \int_{0}^{t} e^{x^{3}} (\frac{x ^{8}}{( x ^{6} + 2 x ^{3} + 2 ) ^{2}}) d x$ . If $f (1) + f^{'} (1) = α e - \frac{1}{6}$ , then the value of $150 α$ is equal to ____ .
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Answer: 16
Q282022 · 30 Jun 2022 · Shift 1· Numerical answer
The probability distribution of X is : $X P (X) 0 \frac{1 - d}{4} 1 \frac{1 + 2 d}{4} 2 \frac{1 - 4 d}{4} 3 \frac{1 + 3 d}{4}$ For the minimum possible value of d, sixty times the mean of X is equal to ____ .
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Answer: 75
Q142022 · 27 Jun 2022 · Shift 1
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. $\frac{1}{34}$
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Answer: C
Q122022 · 27 Jun 2022 · Shift 2
If a point $A (x, y)$ lies in the region bounded by the y-axis, straight lines $2 y + x = 6$ and $5 x - 6 y = 30$ , then the probability that $y < 1$ is :
1. A. $\frac{1}{6}$
2. B. $\frac{5}{6}$
3. C. $\frac{2}{3}$
4. D. $\frac{6}{7}$
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Answer: B
Q192022 · 26 Jun 2022 · Shift 1
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 two heads is :
1. A. $\frac{275}{6 ^{5}}$
2. B. $\frac{36}{5 ^{4}}$
3. C. $\frac{181}{5 ^{5}}$
4. D. $\frac{46}{6 ^{4}}$
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Answer: D
Q72022 · 25 Jun 2022 · Shift 1
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 :
1. A. $P (E_{1} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})$
2. B. $P (E_{1}^{'} \cap E_{2}^{'}) = P (E_{1}^{'}) \cdot P (E_{2})$
3. C. $P (E_{1} \cap E_{2}^{'}) = P (E_{1}) \cdot P (E_{2})$
4. D. $P (E_{1}^{'} \cap E_{2}) = P (E_{1}) \cdot P (E_{2})$
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Answer: C
Q122022 · 25 Jun 2022 · Shift 2
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 the probability, that the sum of the numbers obtained is $48$ , is :
1. A. $\frac{7}{2 ^{11}}$
2. B. $\frac{7}{2 ^{12}}$
3. C. $\frac{3}{2 ^{10}}$
4. D. $\frac{13}{2 ^{12}}$
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Answer: D
Q112022 · 24 Jun 2022 · Shift 2
A random variable X has the following probability distribution : $X P(X) 0 k 1 2 k 2 4 k 3 6 k 4 8 k$ The value of $P (1 < X < 4 ∣ X \leq 2)$ is equal to :
1. A. $\frac{4}{7}$
2. B. $\frac{2}{3}$
3. C. $\frac{3}{7}$
4. D. $\frac{4}{5}$
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Answer: A
Q302022 · 24 Jun 2022 · Shift 2· Numerical answer
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 the remaining 6 questions correctly with probability $\frac{1}{4}$ . If the probability that the student guesses the answers of exactly 8 questions correctly out of 10 is $\frac{27 k}{4 ^{10}}$ , then k is equal to ____.
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Answer: 479