Probability

•Formal study of uncertainty•The engine that drives statistics• Primary objective of lecture unit 4: use the rules of probability to calculate appropriate measures of uncertainty.

Introduction

• Nothing in life is certain• We gauge the chances of

successful outcomes in business, medicine, weather, and other everyday situations such as the lottery (recall the birthday problem)

History

• For most of human history, probability, the formal study of the laws of chance, has been used for only one thing: gambling

History (cont.)• Nobody knows exactly when

gambling began; goes back at least as far as ancient Egypt where 4-sided “astragali” (made from animal heelbones) were used

History (cont.)• The Roman emperor Claudius

(10BC-54AD) wrote the first known treatise on gambling.

• The book “How to Win at Gambling” was lost.

Rule 1: Let Caesar win IVout of V times

Approaches to Probability

• Relative frequencyevent probability = x/n, where x=# of occurrences of event of interest, n=total # of observations

• Coin, die tossing; nuclear power plants?

• Limitationsrepeated observations not practical

Approaches to Probability (cont.)

• Subjective probabilityindividual assigns prob. based on personal experience, anecdotal evidence, etc.

• Classical approachevery possible outcome has equal probability (more later)

Basic Definitions

• Experiment: act or process that leads to a single outcome that cannot be predicted with certainty

• Examples:1. Toss a coin2. Draw 1 card from a standard deck of

cards3. Arrival time of flight from Atlanta to

RDU

Basic Definitions (cont.)

• Sample space: all possible outcomes of an experiment. Denoted by S

• Event: any subset of the sample space S;typically denoted A, B, C, etc.Simple event: event with only 1 outcomeNull event: the empty set Certain event: S

Examples

1. Toss a coin onceS = {H, T}; A = {H}, B = {T} simple events

2. Toss a die once; count dots on upper faceS = {1, 2, 3, 4, 5, 6}A=even # of dots on upper face={2, 4, 6}B=3 or fewer dots on upper face={1, 2, 3}

Laws of Probability

1)(,0)(.2

event any for ,1)(0 1.

SPP

AAP

Laws of Probability (cont.)

3. P(A’ ) = 1 - P(A)For an event A, A’ is the complement of A; A’ is everything in S that is not in A.

AA'

S

Birthday Problem• What is the smallest number of

people you need in a group so that the probability of 2 or more people having the same birthday is greater than 1/2?

• Answer: 23No. of people 23 30 40 60Probability .507.706.891.994

Example: Birthday Problem

• A={at least 2 people in the group have a common birthday}

• A’ = {no one has common birthday}

502.498.1)'(1)(

498.365

343

365

363

365

364)'(

:23365

363

365

364)'(:3

APAPso

AP

people

APpeople

Unions and Intersections

S

A B

A

A

Mutually Exclusive (Disjoint) Events

• Mutually exclusive or disjoint events-no outcomes from S in commonS

AB

A =

Laws of Probability (cont.)

Addition Rule for Disjoint Events:

4. If A and B are disjoint events, then

P(A B) = P(A) + P(B)

Laws of Probability (cont.)

General Addition Rule

5. For any two events A and B

P(A B) = P(A) + P(B) – P(A B)

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

S

A B

A

Example: toss a fair die once

• S = {1, 2, 3, 4, 5, 6}• A = even # appears = {2, 4, 6}• B = 3 or fewer = {1, 2, 3}• P(A B) = P(A) + P(B) - P(A B)

=P({2, 4, 6}) + P({1, 2, 3}) - P({2})

= 3/6 + 3/6 - 1/6 = 5/6

Laws of Probability: Summary

• 1. 0 P(A) 1 for any event A• 2. P() = 0, P(S) = 1• 3. P(A’) = 1 – P(A)• 4. If A and B are disjoint events, then

P(A B) = P(A) + P(B)• 5. For any two events A and B,

P(A B) = P(A) + P(B) – P(A B)

End of First Part of Section 4.1