JNTU
JNTU MCA 2022 PROBABILITY AND STATISTICS Previous Question Paper (May - 2022 - R13)
Answer any five questions. All questions carry equal marks.
Main
Two persons A and B toss a dice. The person who first throws 4 or 5 wins. A starts the game. Show that the probabilities of A’s and B’s winning are in the ratio 3:2.
Answer
Let p be the probability of throwing a 4 or 5 in a single toss of a fair die. Since there are 2 favorable outcomes out of 6, p = 2/6 = 1/3. The probability of not throwing a 4 or 5 is q = 1 - p = 2/3.
A wins if they throw a 4 or 5 on the 1st, 3rd, 5th, ... turn. The probability P(A) is the sum of a geometric series:
Substituting the values:
B wins if they throw a 4 or 5 on the 2nd, 4th, 6th, ... turn. The probability P(B) is:
Substituting the values:
The ratio of the probabilities of A's winning to B's winning is (3/5) : (2/5), which simplifies to 3:2.
Show that for any two events A and B,
Answer
Given the set relations A ∩ B ⊆ A ⊆ A ∪ B, the monotonicity property of probability measures implies P(A ∩ B) ≤ P(A) ≤ P(A ∪ B).
For the final inequality, we use the Addition Theorem of Probability:
Since P(A ∩ B) ≥ 0, subtracting it from P(A) + P(B) results in a value less than or equal to the sum:
Combining these results, we obtain the required inequality chain: P(A ∩ B) ≤ P(A) ≤ P(A ∪ B) ≤ P(A) + P(B).
A, B, C in order toss a coin. The first one to toss head wins the game. What are the probabilities of winning, assuming that the game may continue indefinitely?
Answer
Sign in to view the complete answer paper.
Suppose three companies X, Y, Z produce T.V.’s. X produces twice as many as Y while Y and Z produce the same number. It is known that 2% of X, 2% of Y and 4% of Z are defective. All the T.V.’s produced are put into one shop and then one T.V. is chosen at random. Suppose a T.V. chosen is defective, what is the probability that this T.V. is produced by company X?
Answer
Sign in to view the complete answer paper.
Show that for normal distribution the quartile deviation, mean deviation and standard deviation are approximately 10:12:15.
Answer
Sign in to view the complete answer paper.
The probability of a man hitting a target is 1/3.
i) if he fires 5 times what is the probability of his hitting the target at least twice.
ii) How many times must he fires so that the probability of his hitting the target at least once is more than 90%?
Answer
Sign in to view the complete answer paper.
A manufacturer of pins knows that 2% of his product is defective. If he sells pins in boxes of 100 and guarantees that not more than 4 pins will be defective. What is the probability that a box will fail to meet the guaranteed quality.
Answer
Sign in to view the complete answer paper.
Construct 95% confidence interval for the true proportion of computer literates if 47 out of 150 persons from rural areas are computer literates.
Answer
Sign in to view the complete answer paper.
Write the properties of a good estimator.
Answer
Sign in to view the complete answer paper.
A professor’s feeling about the mean mark in the final examination in probability of a large group of students expressed subjectively by normal distribution with
a) If the mean mark lies in the interval (65, 70) determine the prior probability the professor should assign to the mean mark.
b) Find the posterior mean
c) Determine the posterior probability which he will thus assign to mean mark being in the interval (65, 70) using results obtained in (b).
d) Construct a 95% Bayesian interval for
Answer
Sign in to view the complete answer paper.
In a random sample of 400 persons from a large population, 120 are females. Can it be said that males and females are in the ratio 5:3 in the population? Use 0.05% LOS.
Answer
Sign in to view the complete answer paper.
The owner of a machine shop must decide which of two snack vending machines to install in his shop. If each is tested 250 times, the first machine fails to work 13 times and the second machine fails to work 7 times. Test at the 0.05 level of significance whether the difference between the corresponding sample proportions is significant.
Answer
Sign in to view the complete answer paper.
Heights of fathers and sons are given in centimeters.
| Heights of fathers(x) | 150 | 152 | 155 | 157 | 160 | 161 | 164 | 166 |
| Heights of sons(y) | 154 | 156 | 158 | 159 | 160 | 162 | 161 | 164 |
Find the two lines of regression and calculate the expected average heights of the son when the height of the father is 154 cm.
Answer
Sign in to view the complete answer paper.