IGNOU PGDAST MST 003 Probability Theory Solved Assignment ENGLISH

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MST 003 (January 2026 - July 2026) - ENGLISH

TUTOR MARKED ASSIGNMENT

MST-003: Probability Theory

Course Code: MST-003
Assignment Code: MST-003/TMA/2026

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are True or False. Give a reason in support of your answer.

(a) If events A, B and C are three mutually exclusive and exhaustive events, then it is possible that , and .

(b) If X is a discrete random variable, then X cannot take countably infinite values.

(c) If X is a standard normal variate, then .

(d) If random variables X and Y follow Bernoulli distributions with parameters 1/2 and 1/3 respectively then the random variable X + Y also follows a Bernoulli distribution with parameter 5/6.

2. If a problem is randomly selected from a particular book, then probability that students A, B and C can solve the problem are 1/2, 1/3 and 1/4 respectively. If a problem is selected randomly from this book and given to the students A, B and C then what is the probability that problem is solved. They solve the problem independently.

3. A factory produces certain type of output by 3 machines. The respective daily production figures are- machine X : 4500 units, machine Y: 2000 units and machine Z: 3500 units. Past experience shows that 2% of the output produced by machine X is defective. The corresponding fractions of defectives for the other two machines are 5 and 10 percent respectively. An item is drawn from the day's production. If the drawn item is found to be defective, what is the probability that it has been produced by machine Y?

4. Find the probability that a third child in a family is the family's second daughter, assuming the male and female are equally probable.

5. Two cards are drawn successively with replacement from a well shuffled pack of 52 cards. Find the expected value for the number of kings.

6. Three unbiased coins are tossed simultaneously. In which of the following cases are the events A and B independent?

(i) A be the event of getting exactly one tail

B be the event of getting exactly one head

(ii) A be the event that first coin shows tail

B be the event that third coin shows head

7. Write any 10 characteristics of normal distribution.

8. Four coins were tossed and number of heads noted. The experiment is repeated 200 times. The number of tosses showing 0, 1, 2, 3 and 4 heads were found distributed as under. Fit a binomial distribution to these observed results assuming that the nature of the coins is not known.

Number of Heads	0	1	2	3	4
Number of Tosses	15	35	90	40	20

 

MST 3 2025 - English

TUTOR MARKED ASSIGNMENT

MST-003: Probability Theory

Course Code: MST-003

Assignment Code: MST-003/TMA/2025

Maximum Marks: 100

Note: All questions are compulsory. Answer in your own words.

1. State whether the following statements are True or False and also give the reason in support of your answer.

(a) Sample space of a (i) random experiment tossing two coins simultaneously and (ii) One coin two times is the same.

(b) Standard deviation of a random variable X may take any real value, i.e. its value lies in the interval(ゆす。,

c) If events  are mutually exclusive and exhaustive then will be greater than 1/2 but less than 1.

(d) If S is sample space of a random experiment and E is an event defined on this sample space then P(SE)-1.

(e) If X is a random variable having range set (0, 1, 2, 3) then the set (xS:X(x)0 is an event having at least one outcome of the random experiment.

2. There are 4 black, 3 blue and 8 red balls in an urn. Three balls are selected one by one without replacement. What is the probability that:

(i) First ball drawn is black, second one is red and third one is blue

(ii) All the three balls are of the same colour.

3. A random 5-card poker hand is dealt from a standard deck of cards. Find the probability (in terms of binomial coefficients) of getting a flush (all 5 cards being of the same suit: do not count a royal flush, which is a flush with an ace, king, queen, jack and 10). (10)

4. Show that

is a valid PMF for a discrete random variable. Also find out its CDF.

5. A group of 100 people are comparing their birthdays (as usual, assume their birthdays are independent and not on February 29, etc.). Find the expected number of pairs of people with the same birthday, and the expected number of days in the year on which at least two of these people were born.

6. Random variable X follows Beta distribution with parameters a 3, b-2 and has pdf f(x12x(1x),0x1 10,otherwise Find (1) CDF of X (ii) P[0

Random variable X follows Beta distribution with parameters a=3, b = 2 and has pdf

Find (1) CDF of X (ii) P(0<X<1/2] (iii) mean and variance of X without using direct formula for mean and variance.

(10)

7. Consider the joint PDF for the type of customer service X (0- telephonic hotline, 1- Email) and of satisfaction score Y (1 unsatisfied, 2 satisfied, 3 very satisfied):

	Y
X	1	2	3
0	0	1/2	1/4
1	1/6	1/12	0

(a) Determine and interpret the marginal distributions of both X and Y.

(b) Calculate the 75% quantile for the marginal distribution of Y.

(c) Determine and interpret the conditional distribution of satisfaction level for X=1.

(d) Are the two variables independent?

(e) Calculate and interpret the covariance of X and Y.

8. State Monty Hall problem and solve it.

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