MCom I Semester Statistical Analysis Probability Study Material notes ( Part 4 )

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MCom I Semester Statistical Analysis Probability Study Material notes ( Part 4 )

MCom I Semester Statistical Analysis Probability Study Material notes ( Part 4 ) : Important formulae for Calculation fo probability at a Glance  Long Answer Questions Short Answer Questions Numericals Objective Questions :

Probability Study Material
Probability Study Material

MCom I Semester Statistical Analysis Interpolation Extrapolation Study Material Notes

Illustration 52.

(i) An urn contains 1 black and 2 white balls; another urn contains 2 black and 1 white ball. One ball is taken out at random from the first urn and placed in the second urn and then a ball is drawn at random from the second urn. Prove that the chance of its being white is 5/12.

(ii) Urn 1 contains 2 white and 3 black balls and urn II contains 3 white and 4 black balls. One ball is transferred from urn I to urn II; then one ball is transferred from urn II to urn I. Thereafter one ball is taken out from urn I. What is the probability that the ball so drawn will be white ?

(iii) Two cards are randomly drawn from a deck of cards and thrown away. What is the probability of drawing an ace in a single draw from the remaining 50 cards ?

Solution.

(i) The following are the mutually exclusive situations to get a white ball from the second urn:

(a) By transferring a black ball from first urn to the second urn and then drawing a white ball from it :

Since the first urn contains 1 black and 2 white balls, we have the probability of drawing a black ball i.e., p (bla

If this black ball drawn is placed in the second urn, then the second urn contains 2 + li.e., 3 black and 1 white balls.

Hence, the probability of drawing a white ball from second urn ==

Thus the probability of compound event, drawing black ball from the first urn and then drawing a white ball from the second urn ==

EXAMINATION QUESTIONS

Long Answer Theoretical Questions

1 Define probability and explain the importance of this theory in statistics.

2. What are the different schools of thought on the interpretation of “Probability”? How does each schools define probability ? Explain with suitable examples.

3. Give the classical definition of probability and state its limitations.

4. Explain the concept of independent and mutually exclusive events in probability. State theorems of total and compound probability.

5. Define ‘Probability and explain the addition law of probability giving suitable examples.

6. Define probability and explain the laws of addition and multiplication of probability.

7. Explain what do you understand by the term “probability. State and prove the addition and multiplication theorems of probability.

8. (i) Define probability. Is probability always concerned with only one ‘event’ ? Give suitable example.

(ii) Define Event ‘in’ probability. State the theorem of multiplication of probability for two events which are

(a) independent and

(b) not independent.

9. Explain the concepts of independent and mutually exclusive events in probability. State the theorems of total and compound probability.

10. Define ‘Probability and explain the multiplication theorem of probability giving suitable example.

11. Explain the role of Permutation and Combination in the theory of probability

12. State and prove the ‘Addition’ and ‘Multiplication theorems of probability with suitable examples.

13. What is meant by an ‘event’ in the calculation of probability ? Explain the following:

(a) Independent events.

(b) Mutually exclusive events,

(c) Equally likely events,

(d) Complementary events.

14. Define conditional probability. Explain multiplication theorem for the dependent events.

15. State additive and multiplicative rules and ‘Bays’ theorem. Explain their usefulness. The probabilities of n independent events are respectively P1P2… Pr. Find an expression for probability that at least one of the events will happen.

Short Answer Questions

1 Explain the concept of conditional probability.

2. Distinguish between a priori and a posteriori definitions of probability.

3. State and prove ‘Bays theorem on probability.

4. State and prove Bernoulli’s theorem on Probability.

5. Explain the concepts of independent and dependent events in probability.

6. What is conditional probability ? Explain with the help of an example.

7. Explain the multiplication theorem of probability with suitable examples.

8. Explain the three types of probability concepts.

9. Explain the fundamental concepts of probability.

10. Explain the various approaches of probability.

11. Define the following:

(i) Simple event,

(ii) Compound event,

(iii) Mutually Exclusive events,

(iv) Exhaustive events.

12. What is ‘Bays theorem’? Explain with the help of suitable example ?

13. Write short notes on the following:

(a) Mathematical Expectation,

(b) Conditional Probability,

(C) Inverse Probability,

(d) Bernoulli’s Method.

14. Explain clearly the difference between the following concepts :

(a) Mathematical and statistical probability,

(b) Objective and subjective probability,

(c) Simple and compound probability,

(d) Independent and dependent events,

(e) Permutations and combinations,

(14) A prior and a posteriori probability.

15. State the addition theorem of probability

(a) when two events are mutually exclusive and

(b) when they are not mutually exclusive.

 

 

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