1. Probability and Statistics Presented by Carol Dahl
Basic Probability
Schaum’s Outlines of
Probability and Statistics
Chapter 1
Presented by Carol Dahl
Examples by
Claudianus Adjai

2. Outline of Topics Topics Covered: Set & Set Operations Probabilities
Powerful tools for analysis under uncertainty
Topics Covered:
Set & Set Operations
Probabilities
Counting Rule
Conditional Probability
Probability of a Sample
Permutations & Combinations
Independent Events
Binomial
Bayes’ Theorem

3. Sample Sets
Example:
mining company in Chile owns 240 acres of land with
copper (Cu)
gold (Au)
iron (Fe)
minerals locations distributed as follows:

4. Sample Sets
Au-1 Cu Fe
Au-2 U

5. Set and Set Operations Universal set U = Total Acreage (rectangle)
U  Au = Land contains Au (subset of U)
 = An empty set
Fe  Cu = Land contains iron or copper or both
Fe  Cu = Land contains both iron and copper
Complement of Cu (Cu’):
U - Cu = Cu' = Land not contain copper

6. Probabilities Definition: likelihood that something happens
P(U) = < P(X) < 1
Total of Xi mutually exclusive events for i = 1,2,3,…,n
Drill for minerals randomly
U = 240 acres Fe = 40 acres
Cu = 60 acres Au-1 = 10 acres
Au-2 = 10 acres Cu & Fe = 20 acres

7. Probabilities Au-1 Cu Fe Au-2 U (10 acres) (60 acres) 20 (40 acres)

8. Probabilities What is the probability of finding: Fe deposits?
Cu deposits?
Cu and Fe deposits?
Au-1 deposits?
Au-2 deposits?

9. Counting Rule If equally likely outcomes use counting rule:
P(event) = # of items in event  # of total outcomes
U = 200 acres => P(U) = (200)/200 = 1
Fe = 40 acres => P(Fe) = (40)/200 = 1/5
Cu = 60 acres => P(Cu) = (60)/200 = 3/10

10. Counting Rule Au-1 = 10 acres => P(Au-1) = (10)/200 = 1/20
P(Cu only) = (60-20)/200 = 2/10 = 1/5
P(Fe only) = (40-20)/200 = 1/10
Cu & Fe = 20 acres => P(Cu  Fe) = (20)/200 =1/10

11. Subtraction and Addition Rules
Probability of finding nothing:
= 1 – 1/10 – 2/10 –1/10 – 1/10 = 5/10 = 1/2
=> 50%
Probability find copper or iron (addition rule)
P(Fe  Cu) = P(Fe) + P(Cu) - P(Fe  Cu)
= 2/ /10 – 1/10 = 4/10 = 2/5

12. Conditional Probabilities
Conditional Probability
P(Cu | Fe) = P(Cu  Fe) / P(Fe) = (1/10 ) / (2/10) = 1/2
Probability of a sample
Probability of a model given data
(A,B), (A,C), (B,C), (B,A), (C,A), (C,B)

13. Permutations Example: You own three leases (A,B,C) drill two randomly
without replacement
how many ways can you choose 2 from 3
(A,B), (A,C), (B,C), (B,A), ( , ), ( , )
(A,B), (A,C), (B,C), (B,A), (C,A), (C,B)

14. Permutations and Combinations
If order matters choose r from n:
Permutations = n!/(n-r)! = 3!/(3-2)! = 3 * 2 * 1/1
= 6
If order doesn't matters choose r from n:
Combinations = n!/((n-r)!r!) = 3!/((3-2)!2!)
= 3×2/2 = 3

15. Multiplication Rule and Independence
P(S1 ∩ S2) = P(S1|S2) * P(S1)
Independence:
P(S1 ∩ S2) = P(S1) * P(S2)
Example:
Are discovering Fe and Cu independent?
P(Fe ∩ Cu) = 1/10
P(Fe) *P(Cu) = (2/10)*(3/10) = 6/10

16. Implication of independence
P(Cu|Fe) = P(Cu ∩ Fe) / P(Fe)
= (P(Fe)P(Cu)) / P(Fe) = P(Cu)
marginal probability = conditional

17. Independent Events Example:
Russian gas company Gazprom exploring 4 gas fields
one well per field
similar geology – 1/3 chance of success
probability you get a success on the first two wells
success field independent of success in others
P(S1 ∩ S2) = P(S1) *P(S1) = 1/3*1/3 = 1/9

18. Binomial Notation: Probability of success = p trial = n
without replacement
Formula:
p(X = x) = n!/((n-x)!x!)(p)n(1-p)(n-x)

19. Binomial Probability 2 of 4 are successful (S,S,D,D) = 1/3*1/3*2/3*2/3
(S,D,S,D) = 1/3*2/3*2/3*1/3
(D,D,S,S) = 1/3*1/3*2/3*2/3
(S,D,D,S) = 1/3*2/3*2/3*1/3
(D,S,S,D) = 1/3*2/3*1/3*2/3
(D,S,D,S) = 1/3*2/3*1/3*2/3
P(X = 2) = [4!/((4-2)!2!)](1/3)2(2/3)(n-2) = 0.296

20. Bayes’ Theorem Bayes - Making decisions using new sample information.
Example:
Batteries hybrid renewable energy (wind, solar)
Three of your plants build the battery
E1 E2 E3
Two battery types
regular – r
heavy duty – h

21. Bayes’ Theorem Cont. Example: Factory Types Batteries r h
E r 100 h Total 300
E r 150 h Total 200
E r h Total 100
h battery comes back on warrantee
Probability battery from plant E2 = P(E2|h)?

22. Bayes’ Theorem From definition
P(E2|h) = P(h∩E2) => P(h∩E2) = P(E2|h)P(h)
P(h)
but also P(h∩E2) = P(h|E2)P(E2)
Replace in numerator
P(E2|h) = P(h|E2 )P(E2 )

23. Bayes’ Theorem What is P(h) = P(h∩E1) + P(h∩E2 ) + P(h∩E3 ) But
P(h∩E1) = P(h|E1)P(E1)
P(h∩E2) = P(h|E2)P(E2)
P(h∩E3) = P(h|E3)P(E3)

24. Bayes’ Theorem Replace in denominator P(E2|h) = P(h|E2)P(E2)
P(h|E1)P(E1)+P(h|E2)P(E2)+P(h|E3)P(E3)
= (3/4)(1/3)
(1/2)(1/3) + (1/3)(3/4) + (1/6)(1/2)
= 1/2

25. Bayes’ Theorem General formula (h occurs)
P(Ei|h) = P(h|Ei)P(Ei) P(h|E1)P(E1)+P(h|E2)P(E2)+P(h|E3)P(E3)
P(Ei|h) = P(h|Ei)P(Ei)
i(P(h|Ei)P(Ei)

26. Bayesian Econometrics
Econometrics = wedding model and data
Ei = model, h = data
P(model|data) = P(data|model )P(model)
P(data)
Example: yt =  + ei
P(|y) = P(y|)P()
P(y)

27. Prior and Posterior Distributions
P(|y) = P(y|)P()
P(y)
P(y|) = how well data fits given the model
likelihood function
pick model to maximize
P() = prior beliefs
P(y) no model parameters - treat as exogenous
P(|y) = posterior  likelihood*prior