Welcome To B ASIC C ONCEPT O F P ROBABILITY D EPARTMENT OF S TATISTICS D R. M D. M ESBAHUL A LAM, A SSOCIATE P ROFESSOR E-mail: [email protected] Tel:

Welcome To

BASIC CONCEPT OF PROBABILITY

DEPARTMENT OF

STATISTICSDR. MD. MESBAHUL ALAM, ASSOCIATE

PROFESSOR

E-mail: [email protected] Tel: +88-0721-750041/4122

Probability– Models for random phenomena

Phenomena

DeterministicA mathematical model exists that allows perfect prediction of the phenomena’s outcome.

Non-deterministicNo mathematical model exists that allows perfect prediction of the phenomena’s outcome.

Haphazard

Unpredictable outcomes, but in the long-run the outcomes exhibit no statistical regularity.

Random

Unable to predict the outcomes, but in the long-run, the outcomes exhibit statistical regularity.

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Random phenomena– Unable to predict the outcomes, but in the long-

run, the outcomes exhibit statistical regularity.

Examples

1. Tossing a coin – outcomes S ={Head, Tail}

Unable to predict on each toss whether is Head or Tail.

In the long run we can predict that approx 50% of the time heads will occur and approx 50% of the time tails will occur.

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2. Rolling a die – outcomes

S ={ , , , , , }

Unable to predict outcome but in the long run one can determine that each outcome will occur 1/6 of the time.

Use symmetry. Each side is the same. One side should not occur more frequently than another side in the long run. If the die is not balanced this may not be true.

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3. Rolling a two balanced dice – 36 outcomes 5

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Introduction

• Probability is the study of randomness and uncertainty.

• In the early days, probability was associated with games of chance (gambling).

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History

• Games of chance: 300 BC• 1565: first formalizations• 1654: Fermat & Pascal, conditional probability• Reverend Bayes: 1750’s• 1950: Kolmogorov: axiomatic approach• Objectivists vs subjectivists

– (frequentists vs Bayesians)

• Frequentist build one model• Bayesians use all possible models, with priors

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Concerns

• Future: what is the likelihood that a student will get a CS job given his grades?

• Current: what is the likelihood that a person has cancer given his symptoms?

• Past: what is the likelihood that Monte committed suicide?

• Combining evidence.

• Always: Representation & Inference

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Probability versus Statistics Probability is the field of study that makes

statements about what will occur when a sample is drawn from a known population.

Statistics is the field of study that describes how samples are to be obtained and how inferences are to be made about unknown populations.

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Simple Games Involving Probability

Game: A fair die is rolled. If the result is 2, 3, or 4, you win Tk. 100; if it is 5, you win Tk. 200; but if it is 1 or 6, you lose Tk. 300.

Should you play this game?

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Random Experiment

• a random experiment is a process whose outcome is uncertain.

Examples:

• Tossing a coin once or several times

• Picking a card or cards from a deck

• Measuring temperature of patients

• ...

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Assessing Probability (Classical Approach)

Theoretical/Classical probability—based on theory

(a priori understanding of a phenomena)

For example:

• theoretical probability of rolling a 2 on a standard die is 1/6.

• theoretical probability of choosing an ace from a standard deck is 4/52.

• theoretical probability of getting heads on a regular coin is ½.

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The sample Space, S or The sample space, S, for a random phenomena is the set of all possible outcomes.The sample space S may contain

1. A finite number of outcomes.2. A countably infinite number of outcomes, or3. An uncountably infinite number of

outcomes.

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Note. For cases 1 and 2 we can assign probability to every subset of the sample space. But for case 3 we can not do this. This situation will be discussed in the later workshops.

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A countably infinite number of outcomes means that the outcomes are in a one-one correspondence with the positive integers

{1, 2, 3, 4, 5, …}

This means that the outcomes can be labeled with the positive integers.

S = {O1, O2, O3, O4, O5, …}

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A uncountably infinite number of outcomes means that the outcomes can not be put in a one-one correspondence with the positive integers.

Example: A spinner on a circular disc is spun and points at a value x on a circular disc whose circumference is 1.

0.00.1

0.2

0.3

0.40.5

0.6

0.7

0.8

0.9x S = {x | 0 ≤ x <1} = [0,1)

0.0 1.0

[ )S

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Examples: Sample Space

1. Tossing a coin – outcomes S ={Head, Tail}


S ={ , , , , , }

={1, 2, 3, 4, 5, 6}

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S ={ (1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6),

(2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6),

(4, 1), (4, 2), (4, 3), (4, 4), (4, 5), (4, 6),

(5, 1), (5, 2), (5, 3), (5, 4), (5, 5), (5, 6),

(6, 1), (6, 2), (6, 3), (6, 4), (6, 5), (6, 6)}

outcome (x, y),

x = value showing on die 1

y = value showing on die 2

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3. Rolling a two balanced dice – 36 outcomes

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An Event , E

The event, E, is any subset of the sample space, S. i.e. any set of outcomes (not necessarily all outcomes) of the random phenomena.

S

E

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Examples: Event


S ={ , , , , , }

={1, 2, 3, 4, 5, 6}

E = the event that an even number is rolled

= {2, 4, 6}

={ , , }

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E = the event that “sum of rolled point is 7” ={ (6, 1), (5, 2), (4, 3), (3, 4), (3, 5), (1, 6)}

202. Rolling a two balanced dice.

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Sample SpaceThe sample space is the set of all possible outcomes.

Simple EventsThe individual outcomes are called simple events.

EventAn event is any collectionof one or more simple events

Events & Sample Spaces

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Special Events

The Null Event, The empty event -

= { } = the event that contains no outcomes

The Entire Event, The Sample Space - S

S = the event that contains all outcomes

The empty event, , never occurs.

The entire event, S, always occurs.

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Set operations on EventsUnion

AB

A B

BA

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or A B e e A e B

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• {a,b,c}{2,3} = {a,b,c,2,3}

• {2,3,5}{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}

Union Examples

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Intersection

AB

A B

BA

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• {a,b,c}{2,3} = ___

• {2,4,6}{3,4,5} = ______

Intersection Examples

{4}

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Complement

Let A be any event, then the complement of A (denoted by ) defined by:A

A

A

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A e e A

The event occurs if the event A does not occurA

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In problems you will recognize that you are working with:

1. Union if you see the word or,

2. Intersection if you see the word and,

3. Complement if you see the word not.

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Set operations on Events

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Rules from Set Theory• Commutative Laws:

A B = B A, A B = B A• Associative Laws:

(A B) C = A (B C )

(A B) C = A (B C) .• Distributive Laws:

(A B) C = (A C) (B C)

(A B) C = (A C) (B C)

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1. A B A B

2. A B A B

DeMoivre’s laws

=

=

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1. A B A B

2. A B A B

DeMoivre’s laws (in words)

The event A or B does not occur if the event A does not occur andthe event B does not occur

The event A and B does not occur if the event A does not occur orthe event B does not occur

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A A B A B

Another useful rule

=

In wordsThe event A occurs if A occurs and B occurs or A occurs and B doesn’t occur.

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1. A A

Rules involving the empty set, , and the entire event, S.

2. A 3. A S S

4. A S A

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Definition: mutually exclusive

Two events A and B are called mutually exclusive if:

A B

A B

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Help, I’vebeen

disjointed!

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If two events A and B are mutually exclusive then:

A B

1. They have no outcomes in common.They can’t occur at the same time. The outcome of the random experiment can not belong to both A and B.

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We will use the notation (or )A B B A

to mean that A is a subset B. (B is a superset of A.)

. . if then .e A e B i e

BA

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Definition: Subset/ Superset

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Union: 1 2 31

k

i ki

E E E E E

E1

E2

E3

1 2 31

ii

E E E E

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Intersection: 1 2 31

k

i ki

E E E E E

E1

E2

E3

1 2 31

ii

E E E E

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1. i ii i

E E

DeMorgan’s laws

2. i ii i

E E

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Example: Finite uniform probability spaceIn many examples the sample space S = {o1, o2,

o3, … oN} has a finite number, N, of oucomes.

Also each of the outcomes is equally likely (because of symmetry).

Then P[{oi}] = 1/N and for any event E

no. of outcomes in =

total no. of outcomes

n E n E EP E

n S N

: = no. of elements of n A ANote

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Probability: Classical Approach

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Note: with this definition of P[E], i.e.



n E n E EP E

n S N

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Another Example:We are shooting at an archery target with radius R. The bullseye has radius R/4. There are three other rings with width R/4. We shoot at the target until it is hit

R

S = set of all points in the target

= {(x,y)| x2 + y2 ≤ R2}

E, any event is a sub region (subset) of S.

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E, any event is a sub region (subset) of S.

E

2

: =Area E Area E

P EArea S R

Define

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2

2

1416

R

P BullseyeR

2 2

2

3 29 4 54 4

16 16

R RP White ring

R

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Thus this definition of P[E], i.e.

satisfies the properties:

2

=Area E Area E

P EArea S R

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Finite uniform probability space

Many examples fall into this category

1. Finite number of outcomes

2. All outcomes are equally likely

3.



n E n E EP E

n S N

: = no. of elements of n A ANote

To handle problems in case we have to be able to count. Count n(E) and n(S).

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Properties of Probability1. For the null event , P() = 0.

2. For any event A, 0 P(A) 1.

3. For any event A, P(Ac) = 1 - P(A).

4. If A B, then P(A) P(B).

5. For any two events A and B,

P(A B) = P(A) + P(B) - P(A B).

For three events, A, B, and C,

P(ABC) = P(A) + P(B) + P(C) – P(AB) 3. – P(AC) – P(BC) + P(AB C).

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Independent Events

Intuitively, we define independence as:

Two events A and B are independent if the occurrence or non-occurrence of one of the events has no influence on the occurrence or non-occurrence of the other event.

Mathematically, we write define independence as:

Two events A and B are independent if Pr(A B) = Pr(A)Pr(B).

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Example of Independence

Are party ID and vote choice independent in presidential elections?

Suppose Pr(Rep. ID) = .4, Pr(Rep. Vote) = .5, and Pr(Rep. ID Rep. Vote) = .35

To test for independence, we ask whether:

Pr(Rep. ID) * Pr(Rep. Vote) = .35 ?

Substituting into the equations, we find that:

Pr(Rep. ID) * Pr(Rep. Vote) = .4*.5 = .2 .35,

so the events are not independent.

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Independence of Several Events

The events A1, …, An are independent if:

Pr(A1 A2 … An) = Pr(A1)Pr(A2)…Pr(An)

And, this identity must hold for any subset of events.

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Independent Mutually exclusive

• Events A and ~A are mutually exclusive, but they are NOT independent.

• P(A&~A)= 0• P(A)*P(~A) 0

Conceptually, once A has happened, ~A is impossible; thus, they are completely dependent.

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Conditional ProbabilityConditional probabilities allow us to understand how the probability of an

event A changes after it has been learned that some other event B has occurred.

The key concept for thinking about conditional probabilities is that the occurrence of B reshapes the sample space for subsequent events. - That is, we begin with a sample space S- A and B S- The conditional probability of A given that B looks just at the subset of the sample space for B.

S

A

B

Pr(A | B)The conditional probability of A given B is denoted Pr(A | B).

- Importantly, according to Bayesian orthodoxy, all probability distributions are implicitly or explicitly conditioned on the model.

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Conditional Probability Cont.

By definition: If A and B are two events such that Pr(B) > 0, then:

S

A

B

Pr(A | B)

Pr(B)

B) Pr(A B) |Pr(A

Example: What is the Pr(Republican Vote | Republican Identifier)?

Pr(Rep. Vote Rep. Id) = .35 and Pr(Rep ID) = .4

Thus, Pr(Republican Vote | Republican Identifier) = .35 / .4 = .875

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There are four possible outcomes:Red, RedRed, BlueBlue, RedBlue, Blue

We can find the probabilities of the outcomes by using the multiplication rule for dependent events.

• Suppose there are five balls in an urn. Three are red and two are blue. We will select a ball, note the color, and, without replacing the first ball, select a second ball.

Urn Example54

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Factorials

• For counting numbers 1, 2, 3, …

• ! is read “factorial”– So for example, 5! is read “five factorial”

• n! = n * (n-1) * (n-2) * … * 3 * 2 * 1– So for example, 5! = 5 * 4 * 3 * 2 * 1 = 120

• 1! = 1

• 0! = 1

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Permutations

• Permutation: ordered grouping of objects.

• Counting Rule for Permutations

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Combinations

• A combination is a grouping that pays no attention to order.

• Counting Rule for Combinations

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Useful Properties of Conditional Probabilities

Property 1. The Conditional Probability for Independent EventsIf A and B are independent events, then:

Pr(A)Pr(B)

Pr(B)Pr(A)Pr(B)

B) Pr(A B) |Pr(A

Property 2. The Multiplication Rule for Conditional Probabilities

In an experiment involving two non-independent events A and B, the probability that both A and B occurs can be found in the following two ways:

)A|BPr( Pr(A) B) Pr(A

or

)B|Pr(A Pr(B) B) Pr(A

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Which Drug is Better ?

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Simpson’s Paradox: View I

Drug I Drug II

Success 219 1010

Failure 1801 1190

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 10%

Pr(C|B) ~ 50%

Drug II is better than Drug I

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Simpson’s Paradox: View II

Female Patient

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 20%

Pr(C|B) ~ 5%

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Female Patient

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 20%

Pr(C|B) ~ 5%

Male Patient

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 100%

Pr(C|B) ~ 50%

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Female Patient

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 20%

Pr(C|B) ~ 5%

Male Patient

A = {Using Drug I}

B = {Using Drug II}

C = {Drug succeeds}

Pr(C|A) ~ 100%

Pr(C|B) ~ 50%

Drug I is better than Drug II

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Conditional Independence Event A and B are conditionally independent given

C in case

Pr(AB|C)=Pr(A|C)Pr(B|C) A set of events {Ai} is conditionally independent

given C in case

A and B are independent A and B are conditionally independent

Pr( | ) Pr( | )i iiiA C A C

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Conditional Probability and Partitions of a Sample Space

The set of events A1,…,Ak form a partition of a sample space S if: i=1 to k Ai = S.

If the events A1,…,Ak partition S and if B is any other event in S (note that it is impossible for Ai B = for some i), then the events A1 B, A2 B,…,Ak B will form a partition of B.

Thus, B = (A1 B) (A2 B) … (Ak B)

Pr( B ) = i=1 to k Pr( Ai B )

Finally, if Pr( Ai ) > 0 for all i, then:

Pr( B ) = i=1 to k Pr( B | Ai ) Pr( Ai )

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Example of conditional probability and partitions of a sample space

Pr( B ) = i=1 to k Pr( B | Ai ) Pr( Ai )

Example. What is the Probability of a Republican Vote?

Pr( Rep. Vote ) = Pr( Rep. Vote | Rep. ID ) Pr( Rep. ID )

+ Pr( Rep. Vote | Ind. ID ) Pr( Ind. ID )

+ Pr( Rep. Vote | Dem. ID ) Pr( Dem. ID )

Note: the definition for Pr(B) defined above provides the denominator for Bayes’ Theorem.

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Given two events A and B and suppose that Pr(A) > 0. Then

Example:

Bayes’ Rule

Pr(W|R) R R

W 0.7 0.4

W 0.3 0.6

R: It is a rainy day

W: The grass is wet

Pr(R|W) = ?

Pr(R) = 0.8

)Pr(

)Pr()|Pr(

)Pr(

)Pr()|Pr(

A

BBA

A

ABAB

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Bayes’ RuleR R

W 0.7 0.4

W 0.3 0.6

R: It rains

W: The grass is wet

R W

Information

Pr(W|R)

Inference

Pr(R|W)

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Pr( | ) Pr( )Pr( | )

Pr( )

E H HH E

E

Bayes’ RuleR R

W 0.7 0.4

W 0.3 0.6

R: It rains

W: The grass is wet

Hypothesis H Evidence EInformation: Pr(E|H)

Inference: Pr(H|E) PriorLikelihoodPosterior

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Bayes’ Rule: More Complicated Suppose that B1, B2, … Bk form a partition of S:

Suppose that Pr(Bi) > 0 and Pr(A) > 0. Then

; i j iiB B B S

1

1

Pr( | ) Pr( )Pr( | )

Pr( )

Pr( | ) Pr( )

Pr( )

Pr( | ) Pr( )

Pr( ) Pr( | )

i ii

i ik

jj

i ik

j jj

A B BB A

A

A B B

AB

A B B

B A B

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; i j iiB B B S

1

1

Pr( | ) Pr( )Pr( | )

Pr( )

Pr( | ) Pr( )

Pr( )

Pr( | ) Pr( )

Pr( ) Pr( | )

i ii

i ik

jj

i ik

j jj

A B BB A

A

A B B

AB

A B B

B A B

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; i j iiB B B S

1

1

Pr( | ) Pr( )Pr( | )

Pr( )

Pr( | ) Pr( )

Pr( )

Pr( | ) Pr( )

Pr( ) Pr( | )

i ii

i ik

jj

i ik

j jj

A B BB A

A

A B B

AB

A B B

B A B

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A More Complicated ExampleR It rains

W The grass is wet

U People bring umbrella

Pr(UW|R)=Pr(U|R)Pr(W|R)

Pr(UW| R)=Pr(U| R)Pr(W| R)

R

W U

Pr(W|R) R R

W 0.7 0.4

W 0.3 0.6

Pr(U|R) R R

U 0.9 0.2

U 0.1 0.8

Pr(U|W) = ?

Pr(R) = 0.8

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W The grass is wet




R

W U

Pr(W|R) R R

W 0.7 0.4

W 0.3 0.6

Pr(U|R) R R

U 0.9 0.2

U 0.1 0.8

Pr(U|W) = ?

Pr(R) = 0.8

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W The grass is wet




R

W U

Pr(W|R) R R

W 0.7 0.4

W 0.3 0.6

Pr(U|R) R R

U 0.9 0.2

U 0.1 0.8

Pr(U|W) = ?

Pr(R) = 0.8

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Assessing Probability (Frequency Approach)

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Empirical probability—based on empirical data

For example:

• Toss an irregular die (probabilities unknown) 100 times

and find that you get a 2 twenty-five times; empirical

probability of rolling a 2 is 1/4.

• The empirical probability of an Earthquake in Bay Area

by 2032 is 0.62 (based on historical data)

• The empirical probability of a lifetime smoker developing

lung cancer is 15 percent (based on empirical data).

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Relative Frequency Probability:

If some process is repeated a large number of times, n, and if some resulting event E occurs m times, the relative frequency of occurrence of E, m/n will be approximately equal to the probability of E.P(E) = m/N.

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Interpret this result

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Example:

In a certain population, 10% of the people are rich, 5% are famous, and 3% are both rich and famous. A person is randomly selected from this population. What is the chance that the person is

• not rich?

• rich but not famous?

• either rich or famous?

Subjective Probability

Probability measures the confidence that a particular individual has in the truth of a particular proposition.

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Intuitive Development

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Why Axiomatic Approach?

Rectangle

ConceptReal World

Circle

Probability

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What will be the Appropriate Axioms ?

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The Axiomatic “Definition” of Probability

83

©MD.

MESBAHUL

ALAM

2011

Anomaly in Uncountable Set

• Previous plan is OK when S is finite or countably infinite (discrete)

• For uncountably infinite S additivity makes nonsense– Union of infinite events with probability 0 can make

event of probability one; This makes analysis tough– E.g., for [0,1] as S

• Each singletons is of probability 0• Uncountable union of singletons constitute S which is of

probability 1

84

©MD.

MESBAHUL

ALAM

2011

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