LESSON ONEDECISION ANALYSIS

Subtopic 2 – Basic Concepts from Statistics

Basics of Statistics

Created by The North Carolina School of Science and Math for North Carolina Department of Public Instruction.

Today’s Menu

• Probability

• Expected Value

• Time and Discounting

Basics ofStatistics

Basics: Probability

• What is probability? 3 philosophies

• How do people talk about probability?

By Liam Quin, Licensed CC-BY-3.0, via Wikimedia Commons. http://upload.wikimedia.org/wikipedia/commons/5/59/Five_ivory_dice.jpg

History: Probability First book on probability Modern probability math

Christiaan Huygens Andrey Kolmogorov

(Dutch, 1629-1695) (Russian, 1903-1987)

Axioms of Probability

• Also known as Kolmogorov Axioms

• AXIOM 1 - Probabilities cannot be negative.

• AXIOM 2 - The probability of the set of all possible outcomes is equal to one.

• AXIOM 3 - The probability of a collection of mutually exclusive events is the sum of the individual probabilities of those events.

Axioms of Probability

Axioms of Probability

Axioms of Probability

“or”

Axioms of Probability

• Also known as Kolmogorov Axioms

• AXIOM 1 - Probabilities cannot be negative.

• AXIOM 2 - The probability of the set of all possible outcomes is equal to one.

• AXIOM 3 - The probability of a collection of mutually exclusive events is the sum of the individual probabilities of those events.

Example

Conditional Probability

Independence

Conditional Probability

“and”

Conditional probability example

• Let E1 = {outcome is odd} and E2 = {outcome is

6}. Find P(E2|E1). Find P(E2|not E1).

“and”

Conditional probability example

• Let E1 = {outcome is odd} and E2 = {outcome is

6}. P(E2|E1) = 0/(1/2) = 0. P(E2|not E1)

• = (1/6)/(1/2) = 1/3

“and”

Conditional Probability

“and”

Important note• How to assign probabilities to events is a topic in

statistics (and philosophy).

• Regardless of the method (event space, relative

frequency, or subjective) that generated those

probabilities, once we believe them, the math for

using probabilities in decision making is

always the same.

Beliefs!

• Let B() be a belief function that assigns numbers

to statements such that the higher the number, the

stronger is the degree of belief.

• Beliefs are directly related to probabilities!

• If something is more probable, beliefs that it is

true are stronger than if it is less probable.

Beliefs and Axioms

• Examples: Let F, G, H be events

• Interpret: B(F) > B(G) and B(F|H) > B(G|H)

• Turns out that belief functions

can be constructed out of the

probability axioms.

• Experimentally, we can infer

beliefs by analyzing bets.

Expected value

Expected value examples

• Find the expected value of the face numbers on

one toss of a fair die.

Expected value examples• Find the expected value of the face numbers on

one toss of a fair die. Answer: X1 = 1, X2 = 2,

…, X6 = 6. All have probability 1/6 (fair die).

E(X) = 1(1/6)+2(1/6)

+ 3(1/6) + 4(1/6) + 5(1/6)

+ 6(1/6) = 3.5

Expected value examplesSuppose the prize for beating a chess grandmaster is

$2000, but you have to pay $5 for the opportunity to play

against him. Imagine you’re good at chess, but

not great, so you think it’s only 0.8%

(0.008) likely that you’ll beat him.

Who here would take those odds?

Expected value examplesSuppose the prize for beating a chess grandmaster is

$2000, but you have to pay $5 for the opportunity to play

against him. Imagine you’re good at chess, but

not great, so you think it’s only 0.8%

likely that you’ll beat him. What

is your expected profit/loss from

challenging him?

Expected value examplesX1(lose) = -$5; P(X1) = 0.992

X2(win) = $1995; P(X2) = 0.008

E(X) = X1*P(X1)+X2*P(X2)

= -$5*0.992 + $1995*0.008

= $11.00

Expected value

Discounting• Given an interest rate i = 0.03 (3%) per annum compounded annually, which is the best deal? Let’s guess by show of hands!

• A) $100 000 right now

• B) $104 000 in 18 months

• C) $117 000 in 5 years

• D) $152 000 in 15 years

Discounting

But they’re all at

different points

in time!

What to do??

Discounting

Trick to figuring it out:

Move all of the values to the

same point in time

Discounting• Formula:

• i - interest rate

• n - number of compounding periods

• PV - present value, or value at n = 0

• FV - future value, or value at some n > 0

Discounting• Given an interest rate i = 0.03 (3%) per annum compounded annually, which is the best deal?

• A) $100 000 right now

• B) $104 000 in 18 months

• C) $117 000 in 5 years

• D) $152 000 in 15 years

Solutions

A) PV is given: $100 000

Solutions

B) FV = $104 000, n = 1.5, i = 0.03

Therefore, PV = $99 489.56

Solutions

C) FV = $117 000, n = 5, i = 0.03

Therefore, PV = $100 925.22

Solutions

D) FV = $152 000, n = 15, i = 0.03

Therefore, PV = $97 563.02

Solutions

Best deal is (C), which gives the highest PV.

Discussion: Applications• Which spheres of human endeavor can the

study of decision-making inform?

• What would you guess are some academic

topics being studied in this area?

• What are some questions related to decision-

making that you find interesting?

Homework 1

• Aim: practice using the concepts from this

lesson.