LESSON ONE DECISION ANALYSIS Subtopic 2 – Basic Concepts from Statistics Created by The North Carolina School of Science and Math forThe North Carolina School of Science and Math North Carolina Department of Public Instruction.North Carolina Department of Public Instruction Created by The North Carolina School of Science and Math forThe North Carolina School of Science and Math North Carolina Department of Public Instruction.North Carolina Department of Public Instruction

Today’s Menu Probability Expected Value Time and Discounting

Basics: Probability What is probability? 3 philosophies How do people talk about probability? By Liam Quin, Licensed CC-BY-3.0, via Wikimedia Commons.

History: Probability First book on probability Modern probability math Christiaan Huygens Andrey Kolmogorov (Dutch, ) (Russian, )

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

“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 examples Suppose 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 examples Suppose 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 examples X1(lose) = -$5; P(X1) = X2(win) = $1995; P(X2) = E(X) = X1*P(X1)+X2*P(X2) = -$5* $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) $ right now B) $ in 18 months C) $ in 5 years D) $ 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) $ right now B) $ in 18 months C) $ in 5 years D) $ in 15 years

Solutions A) PV is given: $

Solutions B) FV = $ , n = 1.5, i = 0.03 Therefore, PV = $

Solutions C) FV = $ , n = 5, i = 0.03 Therefore, PV = $

Solutions D) FV = $ , n = 15, i = 0.03 Therefore, PV = $

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.