Sign in
12thPass logo
New chatPYQ LibraryDoubtsRank report
Sign in to see Recents

Your guest activity stays on this device

Sign in to save progress →
Sign in

Probability question

2021 · 26 Aug · Shift 1 · Q25
Guest · filters and generic practice availableBrowsing as a guest · PYQ filters and generic practice are available. Sign in only for personalised features and saved progress.
  1. PYQ Library
  2. /JEE Main
  3. /Mathematics
  4. /Probability
  5. /2021 · 26 Aug · Shift 1 · Q25

Probability question

2021 · 26 Aug · Shift 1 · Q25

JEE MainMathematicsProbabilityMCQ+4 / −1
Let A and B be independent events such that P(A) = p, P(B) = 2p. The largest value of p, for which P (exactly one of A, B occurs) = 59{5 \over 9}95​, is :
  1. A
    13{1 \over 3}31​
  2. B
    29{2 \over 9}92​
  3. C
    49{4 \over 9}94​
  4. D
    512{5 \over 12}125​
View written solutionFree

Correct answer: D

  1. Since AAA and BBB are independent, P(A∩B)=P(A)P(B)=p⋅2p=2p2.P(A\cap B)=P(A)P(B)=p\cdot 2p=2p^2.P(A∩B)=P(A)P(B)=p⋅2p=2p2.

  2. Probability that exactly one of A,BA,BA,B occurs is P(A∩B′)+P(A′∩B).P(A\cap B')+P(A'\cap B).P(A∩B′)+P(A′∩B). Using independence, P(A∩B′)=P(A)−P(A∩B)=p−2p2,P(A\cap B')=P(A)-P(A\cap B)=p-2p^2,P(A∩B′)=P(A)−P(A∩B)=p−2p2, P(A′∩B)=P(B)−P(A∩B)=2p−2p2.P(A'\cap B)=P(B)-P(A\cap B)=2p-2p^2.P(A′∩B)=P(B)−P(A∩B)=2p−2p2.

  3. Therefore, P(exactly one of A,B)=(p−2p2)+(2p−2p2)=3p−4p2.P(\text{exactly one of }A,B)= (p-2p^2)+(2p-2p^2)=3p-4p^2.P(exactly one of A,B)=(p−2p2)+(2p−2p2)=3p−4p2.

  4. Given this equals 59\dfrac{5}{9}95​: 3p−4p2=59.3p-4p^2=\frac{5}{9}.3p−4p2=95​. Rearranging, 4p2−3p+59=0.4p^2-3p+\frac{5}{9}=0.4p2−3p+95​=0. Multiply by 999: 36p2−27p+5=0.36p^2-27p+5=0.36p2−27p+5=0.

  5. Solve the quadratic: p=27±(−27)2−4⋅36⋅52⋅36p=\frac{27\pm\sqrt{(-27)^2-4\cdot 36\cdot 5}}{2\cdot 36}p=2⋅3627±(−27)2−4⋅36⋅5​​ =27±729−72072=\frac{27\pm\sqrt{729-720}}{72}=7227±729−720​​ =27±372.=\frac{27\pm 3}{72}.=7227±3​.

    So, p=3072=512p=\frac{30}{72}=\frac{5}{12}p=7230​=125​ or p=2472=13.p=\frac{24}{72}=\frac{1}{3}.p=7224​=31​.

  6. We need the largest value of ppp, so p=512.p=\frac{5}{12}.p=125​.

  7. Check validity: Since P(B)=2pP(B)=2pP(B)=2p, we need 2p≤1⇒p≤122p\le 1\Rightarrow p\le \frac122p≤1⇒p≤21​. Both values satisfy this, and the larger one is 512\frac{5}{12}125​.

Hence the correct option is D.

PreviousNext

More from Probability

  • 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 ≥ 5 | x > 2) is :2021 · MCQ
  • Two fair dice are thrown. The numbers on them are taken as λ and μ, and a system of linear equations x + y + z = 5 x + 2y + 3z = μ x + 3y +λ z = 1 is constructed. If p is the probability that the system has a unique…2021 · MCQ
  • 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 :2021 · MCQ
  • When a certain biased die is rolled, a particular face occurs with probability 61​−x and its opposite face occurs with probability 61​+x. All other faces occur with probability 61​. Note that opposite…2021 · MCQ
  • The probability distribution of random variable X is given by : Let p = P(1 < X < 4 | X < 3). If 5p = λ K, then λ equal to ​. Includes table2021 · Numerical
  • The probability that a randomly selected 2-digit number belongs to the set {n ∈ N : (2n − 2) is a multiple of 3} is equal to :2021 · MCQ
  • 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 and that of the second unit is 0.8. The instrument is switched on and…2021 · Numerical
  • 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 :2021 · MCQ