STAT 111 Recitation 1 Linjun Zhang January 20, 2017
What s in the recitation This class, and the exam of this class, is a mix of statistical concepts and calculations. We are going to do a little bit of both in recitations: Review the statistical concepts/definitions; Work on practice problems. 1
Miscellaneous Homework 2 is due on next Friday (Jan. 27). The slides of recitations are available on http://stat.wharton.upenn.edu/~linjunz My email address is linjunz@wharton.upenn.edu, and you are very welcome to send me your feedback about the recitations. 2
Statistics are everywhere 3
Statistics draw conclusions from data 4
Probability Theory and Statistics Statistics Inductive Starts with data Makes some inference about reality Probability Theory Deductive Starts with assumptions, If... or Assuming that... Calculates probabilities associated with possible data 5
Probability Theory 6
Events Events: Something that does (or does not) happen once some experiment is performed. We denote events by upper-case letters at the beginning of the alphabet (A, B,...). Experiment 1. Roll a six-sided die. Event A: An even number turns up. Experiment 2. Chose a single ball from a jar that contains 1 red, 3 green, and 2 blue balls. Event A: Get a red ball. Event B: Get either a blue or a green ball. 7
Unions, intersections and complements of events Suppose there are two events D, E. Unions of events: If D and E are events, the union of D and E, written D E, is the event that either D, or E, or both, occur. Intersections of events: If D and E are events, the intersection of D and E, written D E, is the event that both D and E occur. Complements of events: If D is an event, D c is the complement of the event D. Example: Roll a die. D = {even number turn up}, E = {1, 2, 3 turn up}. 8
Probability We write the probability of the event A as Prob(A). Probability is a number between 0 (impossible) and 1 (certain). 9
Unions, intersections and complements of events Suppose there are two events D, E. 10
Mutually exclusive events Two events D and E are mutually exclusive if they cannot both occur together. Then their intersection is the empty event and therefore, from the above equation, if D and E are mutually exclusive, Prob(D E) = Prob(D) + Prob(E). 11
Independence of Events Two events D, E are said to be independent if and only if Prob(D E) = Prob(D) Prob(E). Intuitively, if two events are independent, the knowledge that one of them has occurred does not change the probability that the other occurs. You toss a coin and it comes up heads three times... what is the chance that the next toss will also be a head? 12
Conditional Probabilities Conditional probability: The probability that the event D, occurs, given that the event E occurs is defined as Prob(D E) = Prob(D E). Prob(E) Properties If two events D and E are independent, then Prob(D E) = Prob(D). 13
Concepts of the week The relationship of probability and statistics; 14
Concepts of the week The relationship of probability and statistics; Events (A,B...); 14
Concepts of the week The relationship of probability and statistics; Events (A,B...); Probabilities of events (Prob(A), Prob(B)); 14
Concepts of the week The relationship of probability and statistics; Events (A,B...); Probabilities of events (Prob(A), Prob(B)); Intersections and unions of events (Prob(A B), Prob(A B)); Formula P(A B) = P(A) + P(B) P(A B). 14
Concepts of the week The relationship of probability and statistics; Events (A,B...); Probabilities of events (Prob(A), Prob(B)); Intersections and unions of events (Prob(A B), Prob(A B)); Independence of events and conditional probabilities. 14
Practice problems 1. A coin with probability 0.6 for heads is flipped twice and you are told that at least one head appeared. What is the probability that both flips gave heads? 2. A fair coin was flipped three times and you are told that there was at least one head. What is the probability that all three flips gave heads? 15
Practice problems 1. A coin with probability 0.6 for heads is flipped twice and you are told that at least one head appeared. What is the probability that both flips gave heads? 2. A fair coin was flipped three times and you are told that there was at least one head. What is the probability that all three flips gave heads? (3/7; 1/7) 15
Practice problems 1. A coin with probability 0.6 for heads is flipped twice and you are told that at least one head appeared. What is the probability that both flips gave heads? 2. A fair coin was flipped three times and you are told that there was at least one head. What is the probability that all three flips gave heads? Know where you get stuck, and please do it again tomorrow. 15