1-1

Basic Probability

Schaum’s Outlines of

Probability and Statistics

Chapter 1

Presented by Carol Dahl

Examples by

Claudianus Adjai

1-2Outline of Topics

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

1-3Sample Sets

Example:

mining company in Chile owns 240 acres of land with

copper (Cu)

gold (Au)

iron (Fe)

minerals locations distributed as follows:

1-4

Sample Sets

Cu

Fe

Au-1

Au-2U

1-5Set 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

1-6Probabilities

Definition: likelihood that something happens

P(U) = 1 0 < 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

11

n

iiXP

1-7Probabilities

Cu

(60 acres)

Fe

(40 acres)

Au-1(10 acres)

Au-2(10 acres)(200 acres)

U

20 acres

1-8Probabilities

What is the probability of finding:

Fe deposits?

Cu deposits?

Cu and Fe deposits?

Au-1 deposits?

Au-2 deposits?

1-9Counting 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

1-10Counting Rule

Au-1 = 10 acres => P(Au-1) = (10)/200 = 1/20

Au-2 = 10 acres => P(Au-2) = (10)/200 = 1/20

(Au = 10+10 = 20 acres => P(Au) = 20/200 = 1/10)

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

1-11Subtraction 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 + 3/10 – 1/10 = 4/10 = 2/5

1-12Conditional 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

1-13Permutations

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), ( , ), ( , )

1-14Permutations 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

1-15Multiplication Rule and Independence

Multiplication rule:

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

1-16Implication of independence

P(Cu|Fe) = P(Cu ∩ Fe) / P(Fe)

= (P(Fe)P(Cu)) / P(Fe) = P(Cu)

marginal probability = conditional

1-17Independent 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

1-18Binomial

Notation:

Probability of success = p

trial = n

without replacement

Formula:

p(X = x) = n!/((n-x)!x!)(p)n(1-p)(n-x)

1-19Binomial

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

1-20Bayes’ 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

1-21Bayes’ Theorem

Cont. Example:

Factory Types Batteries

r h

E1 200 r 100 h Total 300

E2 50 r 150 h Total 200

E3 50 r 50 h Total 100

h battery comes back on warrantee

Probability battery from plant E2 = P(E2|h)?

1-22Bayes’ 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 )

P(h)

1-23Bayes’ 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)

1-24Bayes’ 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

1-25Bayes’ 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)

1-26Bayesian 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)

1-27Prior 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