Probability MBA 1st Year Semester Long Question Answers Study Material Notes Study Material Unit Wise Chapter Wise Syllabus Division Of The Content Theory Of Probability, Addition And Multiplication Law, Bays Theorem. Probability theoretical Distributions: Concept/ And Application Of Binomial, Poisson And Normal Distributions.
LONG ANSWER QUESTIONS
Probability MBA 1st Year Semester Long Question Answers Study Material Notes | Index
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Q.1. Describe the addition and multiplication rules of probability with an example for each.(2005-06, 13-14)
Or State addition and multiplication theorems of probability by giving suitable examples. (2014-15)
Ans. Addition Theorem of Probability
We shall discuss the addition theorem of probability for mutually exclusive events and non-mutually exclusive events:
1. Addition Theorem of Probability for Mutually Exclusive Events: If two events A and B are mutually exclusive, then the probability of the occurrence of either A or B is the sum of individ probabilities of A and B, i.e.
2. Equally Likely Events: A set of events is said to be equally likely if one of the events canno expected in preference to any other. For example, if a dice is thrown, any one of the 6 faces numbers 1, 2, 3, 4, 5 and 6 may appear. There is no reason to expect the occurrence of any of the numbers in preference to others. Hence, this set consists of six equally likely events.
3. Mutually Exclusive Events: A set of events is said to be mutually exclusive if the occurrer one of the events includes the possibility of the occurrence of any other. That is at most one of + events can occur.
If a coin is tossed, the two events of appearing of head and tail are mutually exclusive.
If a bag contains several white balls and several black balls and two balls are picked up one after another, the following four events are mutually exclusive, namely:
(a) Picking up a white ball in the first trial and a black ball in the second trial.
(b) Picking up first a black ball and then a white ball.
(c) Picking up both white balls in two trials.
(d) Picking up both black balls in two trials.
4. Exhaustive Events: A set of events is said to be exhaustive if one of the event must occur.
5. Independent and Dependent Events: A set of events is said to be independent if the occurrence of an event is not influenced by the occurrence of any of the other event.
If, on the other hand, the occurrence of one of the event affects the occurrence of the others, the events of the set are said to be dependent.
For example, let the event be that of drawing two aces in two successive trials from a pack of 52 cards.
If the card drawn in the first trial is again placed in the pack, then the two trials will be independent, But if the card is not again placed in the pack, the second trial will be influenced by the first trial, for there remains only 51 cards for the second trial against 52 in the first trial.
6. Algebra of Events: In a random experiment, let’s be some space.
Let A c Sand B SS. Then we observe that:
(a) (An B) is an event that occurs only when each one of A and B occurs.
(b) (AUB) is an event that occurs only when A or B occurs or both occurs.
(c) A is an event that occurs only when A does not occur.
For example; if a bag contains 50 balls and one ball is drawn from it and it is not replaced back and again a second ball is drawn, the drawing of the second ball is dependent on the first ball. However, if the ball is replaced after the first draw, the second drawing will be independent. Hence, two events cannot be mutually exclusive and independent simultaneously. From a pack of playing cards. Ris event drawing a red and B of drawing black. Then, in drawing a card, both R and B can’t happen simultaneously. So these are mutually exclusive events but not independent.
Q.3. State and prove addition theorem of probability.(2006-07)
Or Write short note on addition and multiplication theorem of probability.
Ans. Addition Theorem of Probability: The addition theorem states that if two events A and B are mutually exclusive, the probability of the occurrence of either A or B is the sum of the individual probability of A and B.
I.e. P(A or B) = P(A) + P(B)
Or P(AUB)=P(A) + P(B)
Knowledge of permutation and combination is externally useful in calculating probabilities.
Proof: Ifan event A can happen ina, ways and Binaz ways, then the total number of possibilities is n, then by the definition, the probability of either the first or the second event happening is
Expected frequencies are computed by mathematical methods on the basis of probability theory and it is done even before the investigation is undertaken. For example, if a red colour card is to be drawn from a pack of cards, its chances are % because out of total 52 cards, 26 cards are of red of deviation between theoretical frequencies and actual frequencies, important conclusions are drawn on the basis of various statistical methods.
Utility or Importance of Theoretical Frequency Distribution
Theoretical frequency distribution is the base of modern statistics. Its importance may be placed Under the following heads:
1. Estimation of Nature and Trend of Frequency Distribution: frequency distribution, the nature and trend of frequency distribution can be estimated under certain assumptions and conditions.
2 Basis of Logical Decisions: The phenomenon of risk and uncertainty can be analysed on the basis of theoretical distribution and such analysis is proved very useful in taking logical decisions.
3. Forecasting: Theoretical frequency distribution provides base for prediction, projection and forecasting
4. Substitute of Actual Data: They can be used as substitutes for actual distribution, where to obtain the later is costly or cannot be obtained at all.
5. Test of Sampling: Theoretical frequency distribution serve as benchmarks against which to compare the actual frequency distribution and to find out whether the difference is significant or is merely due to fluctuations of sampling.
6. Solution of Various Problems of Daily Life: Theoretical frequency distribution is of great help in a large number of problems in practical life. For example, a readymade garments manufacturer decides the quantities of various sizes on the basis of normal distribution. In quality control, it can be observed on the basis of Poisson distribution whether the process is in control or not. Similarly, a market researcher can use chi-square distribution to find out the changes in reactions and behaviour of consumers after the change in the nature of the product.
To sum up in the words of Merrill and Fox, they serve as benchmarks against which to compare observed distributions and act as substitutes for actual distribution when the latter are costly to obtain or cannot be obtained at all. They provide decision-makers with a logical basis for making decisions and are useful in making predictions on the basis of limited information or theoretical considerations. In short, theoretical frequency distributions play many important roles in statistical theory.