Jawaharlal Nehru Technological University Hyderabad
Jawaharlal Nehru Technological University Hyderabad MCA 2020 PROBABILITY AND STATISTICS Previous Question Paper (January - 2020 - R09)
Answer any five questions. All questions carry equal marks.
Main
There are 25 cards numbered 1 to 25. Two cards are drawn. Find the probability that their sums is i) even ii) a multiple of 3.
Answer
The total number of ways to draw 2 cards from 25 is 25C2 = 300. Among the 25 cards, there are 13 odd (1, 3, ..., 25) and 12 even (2, 4, ..., 24) numbers.
i) The sum is even if both cards are odd or both are even. Number of ways = 13C2 + 12C2 = 78 + 66 = 144. Probability = 144/300 = 6/25 = 0.24.
ii) For the sum to be a multiple of 3, we consider the remainders when divided by 3: Remainder 0 (3, 6, ..., 24) has 8 cards; Remainder 1 (1, 4, ..., 25) has 9 cards; Remainder 2 (2, 5, ..., 23) has 8 cards. The sum is a multiple of 3 if: (R0, R0) = 8C2 = 28 ways; (R1, R2) = 9 × 8 = 72 ways. Total favorable ways = 28 + 72 = 100. Probability = 100/300 = 1/3 ≈ 0.333.
There are three boxes. I contains 10 light bulbs out of which 4 are defective, II contains 6 light bulbs out of which 1 is defective, III contains 8 light bulbs out of which 3 are defective. A box is chosen at random and a bulb is selected. If it is defective then find the probability that it is from: i) Box- I ii) Box-II iii) Box-III.
Answer
Let B1, B2, B3 be the events of choosing boxes I, II, and III respectively, and D be the event of selecting a defective bulb. P(B1) = P(B2) = P(B3) = 1/3. The conditional probabilities of selecting a defective bulb are P(D|B1) = 4/10 = 0.4, P(D|B2) = 1/6, and P(D|B3) = 3/8 = 0.375.
The total probability P(D) = P(B1)P(D|B1) + P(B2)P(D|B2) + P(B3)P(D|B3) = (1/3)(4/10 + 1/6 + 3/8) = (1/3)(48/120 + 20/120 + 45/120) = 113/360.
Using Bayes' Theorem, P(Bi|D) = [P(Bi)P(D|Bi)] / P(D):
i) P(B1|D) = (1/3 × 4/10) / (113/360) = (4/30) / (113/360) = 48/113 ≈ 0.4248.
ii) P(B2|D) = (1/3 × 1/6) / (113/360) = (1/18) / (113/360) = 20/113 ≈ 0.1770.
iii) P(B3|D) = (1/3 × 3/8) / (113/360) = (1/8) / (113/360) = 45/113 ≈ 0.3982.
A random variable X has the following distribution: X values: 1, 2, 3, 4, 5; P(X=x) values: 1/2, 1/4, 1/8, 1/16, 1/16. Find: i) The mean ii) variance iii) P(1 < X < 4).
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The probability of man hitting a target is 1/3. If he fires 6 times, find the probability of hitting: i) At the most 5 times ii) At least 5 times iii) Exactly once.
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Average number of accidents on any day on a national highway is 1.6. Determine the probability that the number of accidents is i) at least one ii) at the most one.
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The marks obtained by 1000 students is normally distributed with mean 78% and Standard deviation 11%. Determine: i) How many get more than 90% ii) How many students get between 75% and 90%.
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A population consists of four numbers 5, 10, 15, 20. Consider all samples of size two which can be taken with replacement from this population. Find: a) The population mean b) The population standard deviation c) The mean of the sampling distribution of means d) Standard deviation of the sampling distribution of means.
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A random sample of 100 items is taken from a population whose standard deviation is 5.1. The mean of the sample is 21.6. Construct 95% confidence interval for the mean.
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A sample of 121 students is found to have a mean weight of 68 kgs. Can this be regarded as a sample from a population with mean weight 75 kgs and standard deviation 31 kgs?
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A simple sample of heights of 6400 Englishmen has a mean of 67.85 and standard deviation of 2.56 inches. While a simple sample of heights of 1600 Australian has a mean of 68.55 and a S.D. of 2.52 inches. Test the significance between the two means.
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400 articles from a factory are examined and 3% are found to be defective. 600 similar articles from a second factory are found to be 2% defective. Test the significance between the differences of two proportions at 5% level.
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Two random samples are drawn from two normal populations are as follows: Sample I: 17, 27, 18, 25, 27, 29, 13, 17; Sample II: 16, 16, 20, 27, 26, 25, 21. Test whether two populations have been drawn from the same normal population.
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The marks obtained by 10 students in Mathematics and Statistics are given below. Find the coefficient of correlation between the two subjects and the regression lines.
| Marks in Maths | 75 | 30 | 60 | 80 | 53 | 35 | 15 | 40 | 38 | 48 |
|---|---|---|---|---|---|---|---|---|---|---|
| Marks in Statistics | 85 | 45 | 54 | 91 | 38 | 63 | 35 | 43 | 45 | 44 |
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