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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.