1. Probability
Probability – Long Run Relative Frequency of an Event Under a Constant Cause System.
Classical Probability Formula
P(A) = # Outcomes which Favor A
Total # Outcomes
(All Outcomes are equally likely)
Personal or Subjective Probability
Probability that I will get an A
Probability that I will get Married this year
Probability that I will Die this year

2. Random Experiment – An Experiment whose Outcomes are
Determined by Probability
Toss a Coin Select an Employee
Roll a Die Inspect an Item
Draw a Card
Elementary Events – Outcomes which cannot be Decomposed into
other Outcomes
Sample Space – A Collection of all possible Outcomes for an
Experiment (Mutually Exclusive, Exhaustive)
Event – Any Event is a Collection of Elementary Events
A(Even Roll) = {2,4,6} B(High Roll) = {5,6}

3. A B A B Compound Events – Conjunction of two Events
Intersection – The Intersection of Events A and B is composed of all
Elementary Events Common to Both A and B and denoted by:
A and B
A ∩ B
A and B = {6}
Union – The Union of Events A and B is composed of those Elementary
Events Common to A Only, those Common to B Only, and those
Common to Both A and B and denoted by: A or B
A U B
A or B = {2,4,5,6} B A B A

4. Mutually Exclusive – 2 Events are Mutually Exclusive if the occurrence of
one Event precludes the occurrence of the other Event
Independence – Two Events are Independent if the occurrence of one
Event has no effect on the occurrence of the other Event, otherwise
they are Dependent.
Complement – The Complement of Event A, denoted by A, is composed
of all outcomes not in A B A

5. Probability Postulates: SS = {e1, e2, e3, e4, e5, e6}
0 ≤ P( ej ) ≤ 1
P(e1) + P(e2) + P(e3) + P(e4) + P(e5) + P(e6) = 1
If A = {ei, ej, ek), then P(A) = P(ei) + P(ej) + P(ek)
Ex: Choose two people from 3 – Women and 2 – Men
Women = {1,2,3} Men = {4,5}
e1 = {1,2} e5 = {2,3} e8 = {3,4} e10 = {4,5}
e2 = {1,3} e6 = {2,4} e9 = {3,5}
e3 = {1,4} e7 = {2,5}
e4 = {1,5}
(2W) = {e1,e2,e5} (2M) = {e10} (WandM) ={e3,4,e6,e7,e8,e9}

6. Mathematical Counting Rules:
M•N Rule – If there are M first outcomes and N second outcomes,
then there are M*N Total outcomes
Roll a Pair of Dice - 6 * 6 = 36 outcomes
License Plate – ABC *26*26*10*10*10 = 17,576,000
12 A *10*10*10*10*10*10 = 26,000,000
Combinations - # of ways of selecting r items from n total items
without regard to order of selection
n! = n(n-1)(n-2)(n-3)…(3)(2)1
Choose 2 people from 5 people -

7. Draw 5 Cards from a 52 Card Deck - 52C5 =

8. Ex: Select 4 Items from 20 Items of which 5 are Defective

9. 2100 4200 700 900 1800 300 Bivariate Frequency Distribution
Gender\Age <35yr yr >50yr
2100
4200
700
900
1800
300
Male
Female
Probability Distribution
Gender\Age <35yr yr >50yr
Male
Female
Marginal Probability –
Joint Probability -

10. Addition Rule – P( A or B ) = P(A) + P(B) – P( A and B )
Conditional Probability – Probability of Event A Given that Event B is Certain

11. Independence – Events A and B are Independent If
P(A|B) = P(A) and P(B|A) = P(B) ; otherwise Dependent _
Complement Rule – P( A ) + P( A ) = 1
P( A ) = P( A ) _
Game - Toss a Coin until you get a Tail
# Tosses
Outcome T HT HHT HHHT HHHHT HHHHHT HHHHHHT
Probability 1/ / / / / / /128
P(X > 1 Toss) = 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + etc
P( X > 1 Toss) = 1 – P( 1 Toss )

12. Multiplication Rule - P( A and B ) = P( A ) * P( B | A )
Probability Formulas: 1) 2) 3) 4) _
P( A ) = P( A )
P( A or B ) = P(A) + P(B) – P( A and B )
P( A and B ) = P( A ) * P( B | A )

13. Mosaic Plot of Categorical Data

14. Survival Plot for Donner Party

15. Example – 4.35 10 20 15 5 30

16. Example – 4.41

17. Bayes Theorem - Revision of Probability with Additional Information
P(A) – Prior Prob of Event (Subjective)
B – Additional Info (Objective)
P(A|B) – Revised Prob of the Event
Bayes Rule
Example: A – Event that You are an A or B Student , P(A) = .60
B – Score >= 80% on 1st Test P(B|A) = P(B|A) = .15
P(A|B) = _

18. Other Solution Methods:
Prior Likelihood Joint Revised
State P(Ai) P(B|Ai) P(Ai)*P(B|Ai) P(Ai|B)
A or B
C or D

19. Example – 4.34
Birthday Match ?
What is the Probability that at least two people in the Room have the
Same Birthday?

20. Birthday Match ?
What is the Probability that at least two people in the Room have the
Same Birthday?
P(Match) = P(No Match)
P(No Match) = (365/365)(364/365)(363/365)(362/365)(361/365)…etc
# People P(No Match) P(Match)