# 14d probability theory

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PROBABILITY THEORY

There are lots of situations where you can't know an outcome for sure. Probability is

used in this case, to analyze the different possibilities, and how likely each of them is.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and

finding expected values.

a. Find the expected payoff for a game of chance. For example, find the expected

winnings from a state lottery ticket or a game at a fast-food restaurant.b. Evaluate and compare strategies on the basis of expected values. For example,

compare a high-deductible versus a low-deductible automobile insurance policy using

various, but reasonable, chances of having a minor or a major accident.

Use probabilities to make fair decisions (e.g., drawing by lots, using a random number

generator).

Analyze decisions and strategies using probability concepts (e.g., product testing, medical

testing, pulling a hockey goalie at the end of a game).

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THREE VIEWS

1 Subjective ViewProbability is a measure of the strength of ones

expectation that an event will occur.

Example: I think there is a 95% likelihood of Exitobeing a success.

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THREE VIEWS

2 Logical (Mathematical) View

The probability of an event, say A, occurring is givenby the number of events favouring A (nA) divided bythe total number of equally likely events (nS) = nA/nS.

Example: The probability of picking the Ace of Spadesfrom a pack of shuffled cards is 1/52.

This is independentof people!

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THREE VIEWS

3 Empirical (Experimentation) View

The probability of an event A occurring, p(A), is avalue approached by the ratio nA/n as the totalnumber of observations, n, approaches infinity.

Example: The probability of occurrence of a planecrash from past data.

The value may beneither subjective

nor logical.

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EXPERIMENTS AND EVENTS

Experiment:A procedure for carrying out a trial in order toobserve an event or outcome.

Event:An observable happening or outcome.

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SIMPLE EVENT & COMPOUND EVENT

Simple eventsEvent E1 observe a 1Event E2 observe a 2Event E3 observe a 3

Event E4 observe a 4

Event E5 observe a 5Event E6 observe a 6

Compound eventsEvent A observe an odd number

Event B observe an even numberEvent C observe a number < 3

A compound event can be decomposed into simpler events.

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EULER DIAGRAMS

SAMPLE POINTRepresentation of

a simple event

SAMPLE SPACESet of all sample

points

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EXAMPLES EULER DIAGRAMS

Simple eventsEvent E1 observe a 1Event E2 observe a 2Event E3 observe a 3Event E4 observe a 4

Event E5 observe a 5Event E6 observe a 6

Compound eventsEvent A observe an odd number

Event B observe an even numberEvent C observe a number < 3

1 Plot the sample space and events A, B, C.

2 Compute the probabilities of events A, B, C.

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SUMMARY OF PROBABILITIES

Rule 1:Probability assigned to an event lies between 0 and 1.

0 p(Ei) 1

Rule 2:Sum of probabilities over the sample space = 1.

p(Ei) = 1

Rule 3:Probability of a sure event = 1.

p(S) = 1

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COMBINED EVENTS

p(A or B) = probability of occurrence of Event A orprobability of occurrence of Event B orprobability of occurrence of Events A & B

Union of two events: use addition rule

p(A and B) = probability of occurrence of Event A andprobability of occurrence of Event B

Intersection of two events: use multiplication rule

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The probability of the union of two events A and B

p(A or B) = p(A) + p(B) p(A and B)

EXAMPLE: Rolling a dieEvent A: Occurrence of an even number (E2, E4, E6)Event B: Occurrence of a number < 5 (E1, E2, E3, E4)

Compute p(A or B) after drawing the Euler diagram.

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EXAMPLE: Rolling a dieEvent A: Occurrence of an even number (E2, E4, E6)Event B: Occurrence of a number < 5 (E1, E2, E3, E4)Compute p(A or B) after drawing the table of events.

Events A and B Event B and Not A

Event A and Not B Not A and Not B

Event A Not Event A

Event B

Not Event B

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COMPLEMENT RULE OF PROBABILITY

The complement of A is Not A

p(Not A) = 1 - p(A)

EXAMPLE: Rolling a dieEvent A: Occurrence of an even number (E2, E4, E6)Event B: Occurrence of a number < 5 (E1, E2, E3, E4)Compute p{Not (A or B)} from the Euler diagram;

from the table of events.

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EVENTS THAT ARE MUTUALLY EXCLUSIVE

Two events that have no sample points in common aresaid to be mutually exclusive.

Example: Draw the Euler diagram for rolling a die with

Event A = occurrence of an odd numberEvent B = occurrence of an even number

Addition rule of probabilityp(A or B) = p(A) + p(B) p(A and B) becomesp(A or B) = p(A) + p(B)

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EVENTS THAT ARE COLLECTIVELY EXHAUSTIVE

Events for which the probability of their union = 1are called collectively exhaustive events.

Example: Rolling a die with

Event A = occurrence of an odd numberEvent B = occurrence of an even number

p(A) + p(B) = 1

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ADDITION RULE OF PROBABILITY: 3 EVENTS

The probability of the union of three events A, B and C

p(A or B or C) = p(A) + p(B) + p(C)- p(A and B) p(A and C) p(B and C)

+ p(A and B and C)

EXAMPLE: Rolling a dieEvent A: Occurrence of an even number (E2, E4, E6)

Event B: Occurrence of a number < 5 (E1, E2, E3, E4)Event C: Occurrence of a number > 2 (E3, E4, E5, E6)Compute p(A or B or C) after drawing the Eulerdiagram.

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MULTIPLICATION RULE OF PROBABILITY(Statistically independent events)

Two events, A and B, are statistically independent if theprobability of one events occurring is unaffected by the

occurrence of the other.Example: Drawing two cards from a pack with replacement

p(A) = p(A|B)where p(A|B) is probability of A occurring given that B hasoccurred.

EXAMPLE: Rolling a die and tossing a coin togetherEvent A: Occurrence of a headEvent B: Occurrence of the number 4Compute p(A) and p(A|B) after drawing the Eulerdiagram. Are the events statistically independent?

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MULTIPLICATION RULE OF PROBABILITY(Statistically dependent events)

Two events, A and B, are statistically dependent if theprobability of one events occurring is affected by the

occurrence of the other.Example: Drawing two cards from a pack without replacement

p(A) p(A|B)where p(A|B) is probability of A occurring given that B hasoccurred.

EXAMPLE: Students in the MBA programmeEvent A: Student having taken a loanEvent B: Student being a juniorCompute p(A), p(B) and p(A|B) after drawing the tableof events see next slide.

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17 students

MULTIPLICATION RULE OF PROBABILITY(Statistically dependent events)

46 students 37 students

23 students

Junior student Senior student

Loantaken

Loannot

taken

EXAMPLE: Students in the MBA programmeEvent A: Student having taken a loanEvent B: Student being a juniorCompute p(A), p(B) and p(A|B) after drawing the table of events.

Total:40

Total:60

Total:63

Total:83

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MULTIPLICATION RULE OF PROBABILITY(Statistically dependent events)

CONDITIONAL PROBABILITY RULE

p(A|B) = p(A and B)/p(B)

p(B|A) = p(A and B)/p(A)Apply this to the previous slide.

MULTIPLICATION RULE OF PROBABILITYGiven two events A and B, the probability ofoccurrence of both A and B jointly is given by:

p(A and B) = p(A) * p(B|A) = p(B) * p(A|B)

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MULTIPLICATION RULE OF PROBABILITYThree events

MULTIPLICATION RULE OF PROBABILITYGiven three events A, B and C, the probability ofoccurrence of A, B and C jointly is given by:

p(A and B and C) = p(A) * p(B|A) * p(C|A and B)

Example: Sampling without replacement

EXERCISE 3 OF HANDOUT

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