Probability Concepts

Basic Statement on ProbabilityProbability is part of our daily lives. Consider some of the following statements- The price of equity shares of company X is expected to increase- It is likely to rain heavily

Probability is the chance that a particular event will occur.

Basic Terminology in ProbabilityExperiment: An activity that results in one and only one outcome out of a set of disjoint outcomes, where an outcome cannot be predicted with certainty

Events: The possible outcomes of an experiment are known as events

Sample Space: It is the set of all possible outcomes of an experiment

Relative Frequency ApproachBased on statistical dataHere, probability is defined as the proportion of times an event occurs in the long run when the conditions are stableAlternatively, probability is defined as the observed relative frequency of an event in a very large number of events

Classical Approach to ProbabilityProbability can be mathematically represented as

Probability for an event A can be represent as

Example Demand for sandwiches at a Fast Food Corner has always been either 50,100, 150, 200, or 250 per day. Over the past 200 days, the frequencies of demand are represented in the following table:

DemandNo. of days504010080150502002025010Total200

Example

DemandFrequencyProbability5040=40/200 = 0.210080=80/200 = 0.415050=50/200 = 0.2520020=20/200 = 0.125010=10/200 = 0.05Total Probability1

Probability AxiomsEach event must have associated with it a probability greater than or equal to zero but less than or equal to 1

The probability of entire sample is 1

The probability of an event that does not occur is equal to 1 minus probability of the event that occurs

Types of EventsMutually exclusiveEvents are said to be mutually exclusive if only one of the events can occur on any one trial. Example: a fair coin toss results in either a heads or a tails.

Collectively ExhaustiveEvents are said to be collectively exhaustive if their union cover all the events within the entire sample spaceExample: a collectively exhaustive list of possible outcomes for a fair coin toss includes heads and tails.

Law of Addition

Mutually Exclusive EventsP (A or B) = P (A) + P (B)Example: P (cricket or hockey) = P (hockey) + P (cricket)

Law of AdditionNot Mutually Exclusive EventsP(A or B) = P(A)+P(B) - P(A and B)Example: P (cricket or hockey) = P (hockey) + P (cricket) P (cricket and hockey)C & H 10

Example Specialized University offers four different graduate degrees: business, education, accounting, and science. Enrollment figures show 25 of their graduate students are in each specialty. Although 50 of the students are female, only 15 are female business majors. If a student is randomly selected from the Universitys registration database:

What is the probability the student is a business or education major?

What is the probability the student is a female or a business major?

Example The probability that the student is a business or education major is mutually exclusive event. Thus: P(Bus or Edu) = P(Bus) + P(Edu) = .25 + .25 = .50

The probability that the student is a female or a business major is not mutually exclusive because the student could be a female business major. Thus: P(Fem or Bus) = P(Fem) + P(Bus) P(Fem and Bus) = .50 + .25 - .15 = .60

DependencyEvents are either statistically independent (the occurrence of one event has no effect on the probability of occurrence of the other), or statistically dependent (the occurrence of one event gives information about the occurrence of the other).

Probabilities : Independent EventsMarginal probability: the probability of an event occurring: P(A)

Joint probability: the probability of multiple, independent events, occurring at the same time: P(AB) = P(A)*P(B)

Probabilities : Independent EventsConditional probability (for independent events): the probability of event B given that event A has occurred:P(B|A) = P(B) or, the probability of event A given that event B has occurred: P(A|B) = P(A)

Independent Event ExampleA bag contains 3 black balls and 7 green balls. We draw a ball from the bucket, replace it, and draw a second ball.P(black ball drawn on first draw)P(B) = (marginal probability)

P(two green balls drawn)P(GG) =

(joint probability for two independent events)

Independent Event ExampleP(black ball drawn on second draw, first draw was green)P(B|G) = (conditional probability)

P(green ball drawn on second draw, first draw was green)P(G|G) = (conditional probability)A bag contains 3 black balls and 7 green balls. We draw a ball from the bucket, replace it, and draw a second ball.

Probabilities : Dependent EventsMarginal probability: probability of an event occurring: P(A)

Conditional probability : The probability of event B given that event A has occurred: P(B|A) = P(AB)/P(A)The probability of event A given that event B has occurred: P(A|B) = P(AB)/P(B)

Joint probability: The probability of multiple events occurring at the same time: P(AB) = P(B|A)*P(A)

Dependent Event ExampleAssume that we have an bag containing 10 balls of the following descriptions:4 are white (W) and lettered (L)2 are white (W) and numbered (N)3 are yellow (Y) and lettered (L)1 is yellow (Y) and numbered (N)Then:P(WL) =P(WN) =P(W) =P(YL) =P(YN) =P(Y) =

Dependent Event ExampleThen:P(Y) = 0.4 - marginal probability P(L|Y) = P(YL)/P(Y)= 0.3/0.4 = 0.75- conditional probability P(W|L) = P(WL)/P(L)= 0.4/0.7 = 0.57 -- conditional probability

We haveP(WL) = 0.4P(WN) = 0.2P(W) = 0.6P(YL) = 0.3 P(YN) = 0.1P(Y) = 0.4

CombinationA Combination is a way of selecting several things out of a groupIt is given by following formula

Ex: How many combinations of 5 are possible in a standard deck of cards.

PermutationA permutation is an ordered combinationIt is calculated by the following formula

Ex: How many colour codes can be made out of 4 flags of different colours.

Bayes TheoremWe can revise prior estimates of probability using Bayes Theorem

Ex: Suppose there are two boxes B1 and B2. B1 contains 4 white balls and 1 black ball while B2 contains 1 white ball and 4 black balls.

Bayes TheoremLet there be an event A, which can happen, only if one of the n mutually exclusive events occur

Bayes Theorem ExampleThree machines producing 40%, 35% and 25% of the total output are known to produce with defective proportion of items as: 0.04, 0.06 and 0.03, respectively. On a particular day, a unit of output is selected at random, and is found to be defective. What is the probability that it was produced by the second machine.

Bayes Theorem Example In a basin area where oil is likely to be found underneath the surface, there are three locations with three different types of earth compositions, say C1, C2 and C3. The probabilities for these three compositions are 0.5,0.3 and 0.2 respectively. Further it has been found from the past experience that after drilling of well at these locations, the probabilities of finding oil is 0.2, 0.4 and 0.3 respectively. Suppose, a well is drilled at a location, and it yields oil, what is the probability that the earth composition was C1.

Chebychevs LemmaAs per Chebychevs Lemma, SD helps to find out the percentage of population lying within limits from the mean of the variable

For any set of data, and for any constant k>1, at least (1-1/k2) of the observations must lie within k s.ds on either side of the mean

Chebychevs LemmaThe probability that a random variable will lie with in k s.ds of its mean is at least (1-1/k2)

Ex: Let the average number of marks obtained by students in an examination be 60% and s.d. of their marks be 5%. Then find what percentage of students have scores between

Probability/Statistical Distributions

Random VariablesA variable is a quantity which changes or varies due to any factorA variable is called as random variable when there is a chance factor associated with itThe set of all possible values of random variable and their associate probabilities is called as probability distribution

Probability DistributionEx: Let x be the number of breakdowns in machinery of a factory in a given week. The probability distribution for x can be given as follows

No. of breakdowns (x)P(x)00.1210.220.2530.340.13

Expected Value of Random VariableThe expected value of a random variable is the weighted average of all possible values that this random variable can take onThe weights used in computing this average correspond to the probabilities

Example of Expected ValueAn accountant of a company is hoping to receive payment from two outstanding accounts during the current month. He estimates that there is 0.6 probability of receiving Rs.15,000/- due from A and 0.75 probability of receiving Rs.40,000/- due from B. What is the expected cash flow from these two accounts.

Binomial DistributionVisualise a practical situation where a trial results in only two outcomes, say Success and Failure. Further the result of one trial does not influence the result of next trial, and the probability of success at each trail is same.Tossing a coin-Head or TailInspection of an item- Defective or Non-defectiveRepayment by a borrower -Regular or not

Binomial DistributionThe conditions for the applicability of Binomial Distribution are(a) There are n independent trials(b) Each trial has only two possible outcomes(c) The probabilities of two outcomes remain constant

Binomial Distribution If x is the random variable representing number of successes, the probability of getting r successes and (n-r) failures, in n trials, is given by the probability function

Binomial DistributionEx: Spectrum light bulb company went on for mass production of colour bulbs before Diwali. Five bulbs, one of each colour, were packed in each of the boxes which were to be sold as a unit. Due to shortage of time, quality was compromised, and it was estimated that 20% of the bulbs were defective.If a customer purchases such a box of bulbs, what is the probability that the box will haveNo defective bulb2 defective bulbsAt least one defective bulbAt most one defective bulbAll defective bulbs

Using Binomial TablesEx: Seven coins are tossed simultaneously. Using binomial distribution find out the probability of obtaining at least five heads.

Poisson DistributionDistribution of rare eventsThe event whose probability of occurrence is very small but the number of trials which could lead to the occurrence of the event are very largeSo for a large n but very small p such that their product is m=np, the Poisson distribution is defined as

Poisson DistributionEx: Suppose, we have a production process of some item that is manufactured in large quantities. We find that, in general, the proportion of defective items is p=0.01. A random sample of 100 items is selected. What is the probability that there are 2 defective items in this sample.

Poisson DistributionEx: Suppose the probability of dialing a wrong number is 0.05. Then, what is the probability of dialing exactly 3 wrong numbers in 100 dials.

Normal DistributionDistribution for continuous random variableMany real life situations can be applicable to Normal DistributionCharacteristics of Normal Distribution areMean is located at the centre of the distributionCurve is symmetrical around meanMean = Median = Mode

Normal DistributionThe probability density function of a random variable X with

Standard Normal DistributionThe units for standard normal distribution curve are denoted by z and are called as z values or z score.Z gives distance from mean in terms of SD

Properties of Normal Distribution

Normal Distribution ExampleFollowing is the data for life of bulbs manufactured by a company.Mean = 1600 hrsS.D. = 30 hrsWhat is the probability that the life of a bulb selected at random will be less than or equal to a certain value less than the mean, say 1550 hrs?What is the proportion or percentage of bulbs having life less than or equal to 1660 hrs?

Normal Distribution Example(c) What is the probability that the life of a bulb selected at random will be between 1550 and 1580 hrs ?(d) What is the probability that the life of a bulb selected at random will be between 1630 and 1680 hrs ?(e) What is the probability that the life of a bulb selected at random will be between 1550 and 1630 hrs?

SkewnessSkewness is a measure of the asymmetry of the probability distribution

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