Friday 18 November 2022

Business Statistics| Probability| Introduction| Theory | Condition Probability| Bayes' Rule| Law | MBA BBA BCom

What is probability?

·         Probability is the basis for inferential statistics.

·         Inferential statistics means taking a sample from a population, computing a statistic on the sample, and inferring from the statistic the value of the corresponding parameter of the population.

Why use probability?

·         The reason for doing is that the value of the parameter is unknown.

·         Because it is unknown, the analyst conducts the inferential process under uncertainty.

·         By applying rules and laws, the analyst can often assign a probability of obtaining the results.

Use

·         Probability is used directly in certain industries and industry applications.

·         For example, the insurance industry uses probability in actual tables to determine the likelihood of certain outcomes in order to set specific rates and coverage.

·         The gaming industry uses probability values to establish changes and payoffs.

·         In comparing the company figures with those of the general population, the courts could study in probability of a company randomly hiring a certain profile of employees is hired.

·         In manufacturing and aerospace, it is important to know the life of mechanized part and the probability that it will malfunction at any given length of time in order to protect the firm from major breakdowns.

Methods of Assigning Probabilities

·         The three general methods of assigning probabilities



1.       Classical method of Assigning probabilities

·         When probabilities are assigned based on laws and rules, the method is referred to as classical method.

·         This method involves an experiment, which is a process that produces outcomes, and an event, which is an outcome of an experiment.

·         Using the classical method, the probability of an individual event occurring is determined as the ratio of the number of items in a population containing the event –r, to the notal number of items in the population- N.

·       Range of possible probabilities 0 P (E) 1.

2.       Relative Frequency of Occurrence

·         In this method assigning probabilities is based on cumulated historical data.

·         With this method, the probability of an event occurring is equal to the number of times the event has occurred in the past divided by the total number of opportunities for the event to have occurred.

·       


·        

Relative frequency of occurrence is not based on rules or laws but on what has occurred in the past.

3.       Subjective Probability

·         This method based on the feelings or insights of person determining the probability.

·         Subjective probability comes from the person’s intuition or reasoning. The subjective method often is based on the accumulation of knowledge, understanding, and experience stored and processed in the human mind.  At times it is merely a guess.

·         Subjective probability can be used to capitalize on the background of experienced workers and managers in decision making.

·         Subjective probability also can be a potentially useful way of tapping a person’s experience, knowledge, and insight and using them to forecast the occurrence of some event.

Structure of Probability

1.       Experiment: an experiment is a process that produces outcomes.

·         Auditing every 10th account to detect any errors.

·         Interviewing 20 randomly selected consumers and asking them which brand of appliance they prefer.

2.       Event: an event is an outcome of an experiment.

·         The experiment defines the possibilities of the event.

·         If the experiment is to sample five bottles coming off a production line, an event could be to get one defective and four good bottles.

3.       Elementary Event

·         Events that cannot be decomposed or broken down into other events are called elementary events.

·         Suppose the experiment is to roll a die. The elementary events for this experiment are to roll a 1 or roll a 2 or roll a3, and so on.

4.       Sample space: sample space is a complete roster or listing of all elementary events for an experiment.

5.       Union and intersections: the union is formed by combining elements from both sets.

·         The intersection contains the elements common to both sets.

6.       Mutually exclusive events:

·         Two or more events are mutually exclusive events

·         If the occurrence of one event precludes the occurrence of the other events.

·         This characteristic means that mutually exclusive events cannot occur simultaneously and therefore can have no intersection.

7.       Independent Events

·         Two or more events are independent events if the occurrence or nonoccurrence of one of the events does not affect the occurrence or nonoccurrence of the other events.

8.       Collectively exhaustive events

·         A list of collectively exhaustive events contains all possible elementary events for an experiment.

·         All sample spaces are collectively exhaustive lists.

9.       Complementary event

·         The complement of event A is denoted A’. It is “not A”.

·         All the elementary events of an experiment not in A comprise its compliant.

·         P(A’) = 1- P(A)

General Law of Addition

·         Where X,Y are events

Special Law of Addition

·         When the events are mutually exclusive, a zero is inserted into the general law formula for the intersection, resulting in the special law formula.

General Law of Multiplication

·         General Law of multiplication gives the probability that both event X and event Y Will occur at the same time.

Special Law of Multiplication

·         If events X and Y are independent, a special law of multiplication can be used to find the intersection of X and Y.

Conditional Probability

·         Conditional probabilities are computed based on the prior knowledge that a statistician has one of the two events being studied.

·         If X, Y are events, the conditional probability of X occurring given that Y is known or has occurred is expressed as P(X|Y) and is given it the law of conditional probability.


Revision of Probabilities: Bayes' Rule

·         An extension to the conditional law of probabilities is Bayes' rule, which was developed by and named for Thomas Bayes (1702-1761).

·         Bayes' rule is a formula that extends the use of the law of conditional probabilities that allow revision of original probabilities with new information.

·         By expressing the law of conditional probabilities in this new way, Bayes' rule enables the statistician to make new and different application using conditional probabilities.

·         In particular, statisticians use Bayes' rule to revise probabilities in light of new information.

·         


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Sunday 30 July 2017

AS 2 SY BBA


AS-2  
  •  Assuming that the typing mistake per page committed by a typist follows a Poisson distribution; find the expected frequencies for the following distribution of typing mistakes:

Number of mistake per page
0
1
2
3
4
5
Number of pages
40
30
20
15
10
5
 (e-1.5 = 0.22313)  
  • The following table give the number of days in a 50 days period during which automobile accidents occurred in a city. Fit a poisson distribution to the data (e-0.9 = 0.4066)

No. of accidents
0
1
2
3
4
No. of days
21
18
7
3
1



  •  It is given that 3 per cent of electric bulbs manufactured by a company are defective. Find the probability that a sample of 100 bulbs will contain (i) no defective (ii) exactly one defective.  (e-3  = 0.05)
  •  In a company during a break time of half an hour on average 15 calls are coming. find the probability that (i) there are at least 2 calls per five minutes (ii) There are at the most 2 calls per five minutes.
  •  If mean of a Poisson distribution is 4, then find standard deviation of the distribution.
  • The minor injuries a cricket player can expect during the net practice is a random variable having the Poission distribution with (Lamda) = 4.4. Find the probability that during the net practice there will be at the most one minor injuries.
  • Give properties of Poission distribution.

Sunday 23 July 2017

SY BBA QM III : AS-1

AS-1


  1. The probability distribution of a random variable x is given below. Find E(x) and V(x).
    X
    -2
    -1
    0
    1
    2
    P(x)
    2k
    K
    3k
    3k
    k
  2.  The following table show the distribution of 128 samples. Fit a binomial distribution and find expected frequency:
X
0
1
2
3
4
5
6
7
Samples
7
6
19
35
30
23
7
1
2.                 
                        3. The probability function of binomial distribution is P(x) = 10Cx (0.8)^x (0.2)^(10-x), then find its variance.
3.       The probability distribution of a random variable  x  is as follow: Find mean and S.D.
Xi
2
3
4
5
6
7
8
9
10
P(Xi)
0.05
0.1
0.3
0.2
0.05
0.1
0.05
0.1
0.05
4.       
                4. The probability distribution of a random variable x is given below, find mean and variance.
X
8
12
16
20
24
P(X)
1/8
1/6
3/8
1/4
1/12
5.       
                         5.    Four coins are tossed simultaneously. find the probability of getting 2 head.
6.       
                6.   A and B play a game in which the probability of winning of A is 2/3, find the probability that A will win at least 6 times out of 8 trials.

               7.  The probability that in certain correspondence course student will graduate is 0.4. Determine the probability that out of 5 students (a)none (b) one (c) at least one will graduate.