Intro to Probability

Intro to Probability Martina Litschmannová martina.litschmannova@vsb.cz K210

Probability basics

Statistical (random) experiment A random experiment is an action or process that leads to one of several possible outcomes. For example:

Statistical (random) experiment All statistical experiments have three things in common: • The experiment can have more than one possible outcome. • Each possible outcome can be specified in advance. • The outcome of the experiment depends on chance.

Probability Terminology • Asample spaceΩ is a set of elements that represents all possible outcomes of a statistical experiment. No two outcomes can occur at the same time! • A simpleevent is an element of a sample space.Simpleevents are denoted by , • An event is a subset of a sample space - one or more sample points.Eventsare denoted by , or

Probability Terminology • Using notation from set theory, we can represent the sample space and its outcomes as .

TypesofEvents • Two events are mutually exclusive if they have no sample points in common. • Two events are independent when the occurrence of one does not affect the probability of the occurrence of the other.

Requirements of Probabilities Given a sample space , the probabilitiesassigned to the outcome must satisfy these requirements: • The probability of any outcome is between 0 and 1, i.e.for each i. • The sum of the probabilities of all the outcomes equals 1, i.e.++…+=1. represents the probability of outcome i

Approaches to Assigning Probabilities There are three ways to assign a probability, , to an outcome, , namely: • Classical approach: make certain assumptions (such as equally likely, independence) about situation. • Relative frequency: assigning probabilities based on experimentation or historical data. • Subjective approach: Assigning probabilities based on the assignor’s judgment.

Classical Approach If an experiment has n possible outcomes(all equally likely to occur), this method would assign a probability of 1/n to each outcome.

You are rolling a fair die. What is the probability that one will fall? Experiment: Rolling a die Sample Space: Probabilities: Each sample point has a 1/6 chance of occurring.

What about randomly selecting a student fromthisclassand observing their gender? Experiment: Randomselectionofa student fromthisclass Sample Space: Probabilities: ??? Are these probabilities ½?

Experiment: Rolling 2 (fair) die (dice)and summing 2 numbers on top. Sample Space: Probability Examples:

Relative Frequency Approach • Computer Shop tracks the number of desktop computer systems it sells over a month (30 days): For example, 10 days out of 30 2 desktops were sold. From this we can constructthe “estimated” probabilities of an event(i.e. the # of desktop sold on a given day).

Relative Frequency Approach…

Relative Frequency Approach… There is a 40% chance ComputerShopwill sell 3 desktops on any given day (Based on estimates obtained from sample of 30 days).

Subjective Approach In the subjective approach we define probability as the degree of belief that we hold in the occurrence of an event. Forexample: • P(you drop this course) • P(NASA successfully land a man on the moon) • P(girlfriend/boyfriend says yes when you ask her to marry you)

Events & Probabilities • An individual outcome of a sample space is called a simple event(cannot break it down into several other events). • An event is a collection or set of one or more simple events in a sample space. Forexample: Roll of a die: • Simple event:The number “3” will be rolled. • Event: An even number (one of 2, 4, or 6) will be rolled.

Events & Probabilities The probability of an event is the sum of the probabilities of the simple events that constitute the event. Forexample: Assuming a fair die, rollof a die: and . Then:

Interpreting Probability One way to interpret probability is this: If a random experiment is repeated an infinite number of times, the relative frequency for any given outcome is the probability of this outcome. For example, the probability of heads in flip of a balanced coin is 0,5, determined using the classical approach. The probability is interpreted as being the long-term relative frequency of heads if the coin is flipped an infinite number of times.

Joint, Marginal and Conditional Probability We study methods to determine probabilities of events that result from combining other events in various ways. There are several types of combinations and relationships between events: • Complement of an event - (everything other than that event) • Intersection of two events- (event A and event B) • Union of two events- (event A or event B)

Why are some mutual fund managers more successful than others? One possible factor is where the manager earned his or her MBA. The following table compares mutual fund performance against the ranking of the school where the fund manager earned their MBA: Where do we get these probabilities from? E.g. This is the probability that a mutual fund outperforms AND the manager was in a top-20 MBA program; it’s a joint probability [intersection].

Alternatively, we could introduce shorthand notation to represent the events:A1 = Fund manager graduated from a top-20 MBA programA2 = Fund manager did not graduate from a top-20 MBA programB1 = Fund outperforms the market B2 = Fund does not outperform the market E.g. = the probability a fund outperforms the marketandthe manager isn’t from a top-20 school.

Marginal Probabilities Marginal probabilities are computed by adding across rows and down columns; that is they are calculated in the margins of the table:

Marginal Probabilities Marginal probabilities are computed by adding across rows and down columns; that is they are calculated in the margins of the table: BOTH margins must add to 1 (useful error check)

Marginal Probabilities Marginal probabilities are computed by adding across rows and down columns; that is they are calculated in the margins of the table: What’s the probability a fundmanager isn’t from a top school?

Marginal Probabilities Marginal probabilities are computed by adding across rows and down columns; that is they are calculated in the margins of the table: What’s the probability a fundoutperforms the market?

Conditional Probability Conditional probability is used to determine how two events are related; that is, we can determine the probability of one event given the occurrence of another related event. Experiment: random select one student in class. Conditional probabilities are written as and read as “the probability of A given B” and is calculated as:

Rule ofMultiplication Again, the probability of an event given that another event has occurred is called a conditional probability. both are true - keep this in mind!

What’s the probability that a fund will outperform the market given that the manager graduated from a top-20 MBA program? Recall: A1 = Fund manager graduated from a top-20 MBA program A2 = Fund manager did not graduate from a top-20 MBA program B1 = Fund outperforms the market B2 = Fund does not outperform the market Thus, we want to know:What is?

What’s the probability that a fund will outperform the market given that the manager graduated from a top-20 MBA program? Thus, there is a 27,5% chance that a fund will outperform the market given that the manager graduated from a top-20 MBA program.

Independence One of the objectives of calculating conditional probability is to determine whether two events are related. In particular, we would like to know whether they are independent, that is, if the probability of one event is not affected by the occurrence of the other event. Two events A and B are said to be independent if and .

Independence For example, we saw that The marginal probability for B1 is: Since B1 and A1 are not independent events.

Union • Determine the probability that a fund outperforms (B1)orthe manager graduated from a top-20 MBA program (A1). A1 or B1 occurs whenever: A1 and B1occurs,A1 and B2occurs, orA2 and B1occurs.

Rule ofaddition • Determine the probability that a fund outperforms (B1)or the manager graduated from a top-20 MBA program (A1).

Probability Rules and Trees We introduce three rules that enable us to calculate the probability of more complex events from the probability of simpler events: • The Complement Rule: (May be easier to calculate the probability of the complement of an event and then substract it from 1,0 to get the probability of the event.) • The Multiplication Rule: • The Addition Rule:

A graduate statistics course has seven male and three female students. The professor wants to select two students at random to help her conduct a research project. What is the probability that the two students chosen are female? • Let F1 represent the event that the first student is female. • Let F2 represent the event that the second student is female. and events are NOT independent!

Intro to Probability