Chapter 5 – Probability Probability is a way of getting a handle on the randomness that exists in the world – it quantifies and describes random events. Understanding probability is the first step on the path toward inferential statistics.Thought Experiment: If you have a fair coin and flip it 10 times and get H all 10 flips, that maynot be too surprising. Flipping it 50, 100, or 1000 times and getting all H would be surprising  we expect randomness and understand that H would occur with low probability in these scenarios.In fact, long-run proportions, such as the proportion of H in 1000 coin flips, provide the basis forunderstanding probability.Definition: the probability of a particular outcome in a randomized experiment or other process is the proportion of times that outcome would occur in a long run of observations.With random processes, the proportion of times an event happens is highly random and variable in the short run, but very predictable in the long run. This is due to the law of large numbers (andassuming independent trials).Definition: different trials of a random process are independent if the outcome of any one trial is not affected by the outcome of any other trial. This process is said to be memoryless.EDefinition: the sample space, S, is the set of all possible outcomes of a random process.Example:1. Flip a coin. What is the sample space? S = {H, T}2. Flip two coins. What is the sample space? S = { }The actual elements in a sample space are called events, which are particular outcomes or groupsof particular outcomes.Events have probabilities. Rules of probability:- The probability of the whole sample space, Pr[S], is 1.- The probability of {∅ } (the empty set) = 0- The sum of the probabilities for each individual event in S is 1.- A probability must be between 0 and 1 (inclusive).For an event A, we define Pr[A] = P(A) = pA = “probability of event A”Pr[A] is the sum of all probabilities of individual outcomes in event A.Example:Roll a fair die. Let A be the event of getting an odd number.What is S?What is A?What is Pr[A]?Definition: the complement of an event A, is all outcomes in the sample space that are not in A. This is denoted AC.Pr[A] + Pr[AC] = Pr[S] = 1Example:If we roll a die and let A be the event of obtaining an odd number, what is AC?Definition: Two events are disjoint, or mutually exclusive, if they don’t have any outcome in common.Example:Roll a die. A is the event of an odd number. B is the event of rolling 4.A = { } B = { }Definition: The intersection of two events is all outcomes they have in common; it is the elements in both events. Denoted ‘∩’Definition: The union of two events is all in either event; it is the elements in at least one event. Denoted ‘∪’Example:Roll a die. A is the event of an odd number. C is the event of a number less than 4.A∩C = { }A∪C = { }How do we find the probabilities of the union and intersection? By using the General Addition Rule.Pr[A ∪B]=Pr[A]+Pr[B]−Pr ⁡[ A ∩B ]If we know two events are independent, we can use the Multiplication Rule:Pr[A]× Pr[B]=Pr ⁡[ A ∩B ]↔ A and B are independent eventsNB: We cannot use the multiplication rule unless A and B are independent, but we *can* use it tocheck for independence. It is not often that we have independent events, so do not count on this occurring.Question: If A and B are disjoint, what is Pr[A∩B]?In Chapter 3 we used conditional proportions to see if two categorical variables were associated,.Conditional probabilities are similar. We look at the probability conditioned on a subsert of the sample space rather than over the whole sample space, like we looked at the proportion of a sample conditioned on a subset of the observations.Definition: the conditional probability of an event is the probability of that event’s occurring when you know that outcome was in some part of the sample space (when you already have some other information about the outcome).- The probability that UGA wins given that they play at homeo A is the event UGA winso B is the event UGA plays at home- The probability you get an odd number given that the number rolled was below 5o A is event of an odd numbero B is event of a number less than 5How do we calculate it? We have to take into account that both events must take place (the intersection), but also the prior information about the outcome (e.g., UGA played at home; the number rolled was less than 5).We do this using Bayes’ Rule.The conditional probability of event A given event B is:Pr[A|B]=Pr[A ∩B]Pr [B]Using this, find the probability of getting an odd number given we know we’ve rolled a number less than 5 (on a fair die).A = {1, 3, 5}; B = {1, 2, 3, 4}A ∩ B = {1, 3}Pr[A∩B] = 2/6; Pr[B] = 4/6Pr[A|B]=Pr[A ∩B]Pr [B]=2/64/6=2 /4Conditioning on an event is like reducing the set of possible outcomes to just those in that event.Example:The probability of getting H on a fair coin is 1/2. You flip the coin twice, but the second time it rolls under the table and you can’t ‘see it. Given that the first flip was H, what is the probability of H on the second?Pr[A|B] = Pr[A] ↔A and B are independent**Read §5.4Understanding probability is the key to inferential statistics. Things are random and variable, but probability tells us about the chance of given observations within a random