JNTU
JNTU MCA I PROBABILITY AND STATISTICS Previous Question Paper (March - 2023 - R17)
Note: i) Question paper consists of Part A, Part B. ii) Part A is compulsory, which carries 25 marks. In Part A, Answer all questions. iii) In Part B, Answer any one question from each unit. Each question carries 10 marks and may have a, b as sub questions.
PART A
Define dependent and independent events. Show that if E is independent of F then E is also independent of Fc (F complement).
Answer
Two events E and F are independent if the occurrence of one does not affect the probability of the other, defined by P(E ∩ F) = P(E)P(F). Otherwise, they are dependent. To show E is independent of Fc, we use the fact that E = (E ∩ F) ∪ (E ∩ Fc), where these are mutually exclusive. Thus, P(E) = P(E ∩ F) + P(E ∩ Fc). Substituting P(E ∩ F) = P(E)P(F), we get P(E ∩ Fc) = P(E) - P(E)P(F) = P(E)(1 - P(F)). Since 1 - P(F) = P(Fc), we have P(E ∩ Fc) = P(E)P(Fc), proving E and Fc are independent.
If X is a normal random variable with mean μ = 3 and variance σ2 = 16, find (i) P(X < 11) (ii) P(X > -1) (iii) P(2 < X < 7).
Answer
Given X ~ N(3, 16), we have μ = 3 and σ = 4. We use the standard normal variable Z = (X - 3) / 4.
- P(X < 11) = P(Z < (11-3)/4) = P(Z < 2) ≈ 0.9772.
- P(X > -1) = P(Z > (-1-3)/4) = P(Z > -1) = P(Z < 1) ≈ 0.8413.
- P(2 < X < 7) = P((2-3)/4 < Z < (7-3)/4) = P(-0.25 < Z < 1) = P(Z < 1) - P(Z < -0.25) = 0.8413 - 0.4013 = 0.4400.
Find the probability that in 120 tosses of a fair coin (i) between 40% and 60% will be heads, (ii) 5/8 or more will be heads.
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An instructor has two classes, A and B. Class A has 16 students, class B has 25 students. Class A had a standard deviation of 9 while class B had a standard deviation of 12. Can we conclude at the (i) 0.01, (ii) 0.05 level of significance that the variability of class B is greater than that of A?
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Derive the normal equations for the fit of the straight line
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PART B
In answering a multiple-choice test, a student either knows the answer or guesses. Let p be the probability she knows and 1-p she guesses. If she guesses, she is correct with probability 1/m, where m is the number of alternatives. What is the conditional probability she knew the answer given she was correct?
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A bin contains 5 defective, 10 partially defective, and 25 acceptable transistors. A transistor is chosen at random and put into use. If it does not immediately fail, what is the probability it is acceptable?
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Events A (Asian project successful) and B (European project successful) are independent with P(A)=0.4 and P(B)=0.7. (i) If A fails, probability B fails? (ii) Prob at least one success? (iii) Given at least one success, prob only A is successful?
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A magazine publishes columns A, B, C. Reading probabilities are provided. Find conditional probabilities based on the provided table data.
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For density function
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Mean weight of 500 students is 151 lb, SD is 15 lb. Assuming normal distribution, find how many weigh (i) between 120 and 155 lb, (ii) more than 185 lb.
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Prove that mean and variance of a binomial distribution are
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If prob of reaction to serum is 0.001, find prob that out of 2000, (i) exactly 3, (ii) more than 2, suffer reaction.
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Population 2, 3, 6, 8, 11. Find population mean, SD, and mean/SD of sampling distribution for samples of size 2 with replacement.
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Sample of 200 ball bearings has mean 0.824 and SD 0.042. Find 95% and 99% confidence limits for mean.
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SD=0.05. Find sample size needed for 95% and 99% confidence that error does not exceed 0.01.
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100 voters, 55% favor candidate. Find 95%, 99%, 99.73% confidence limits for population proportion.
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ESP experiment: Subject identifies 32 cards correctly out of 50. Determine significance at 0.05 and 0.01 levels.
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Machine mean thickness 0.050. Sample of 10 gives mean 0.053, SD 0.003. Test hypothesis at 0.05 and 0.01 levels.
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Calculate the coefficient of correlation and the two lines of regression for data: (x: 10, 12, 18, 24, 23, 27), (y: 13, 18, 12, 25, 30, 10).
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Fit a curve of the form
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