Lecture 7 Dustin Lueker

 Experiment ◦ Any activity from which an outcome, measurement, or other such result is obtained  Random (or Chance) Experiment ◦ An experiment with the property that the outcome cannot be predicted with certainty  Outcome ◦ Any possible result of an experiment  Sample Space ◦ Collection of all possible outcomes of an experiment  Event ◦ A specific collection of outcomes  Simple Event ◦ An event consisting of exactly one outcome 2STA 291 Summer 2008 Lecture 7

 Let A and B denote two events  Complement of A ◦ All the outcomes in the sample space S that do not belong to the even A ◦ P(A c )=1-P(A)  Union of A and B ◦ A ∪ B ◦ All the outcomes in S that belong to at least one of A or B  Intersection of A and B ◦ A ∩ B ◦ All the outcomes in S that belong to both A and B 3STA 291 Summer 2008 Lecture 7

 Let A and B be two events in a sample space S ◦ P(A∪B)=P(A)+P(B)-P(A∩B)  A and B are Disjoint (mutually exclusive) events if there are no outcomes common to both A and B ◦ A∩B=Ø  Ø = empty set or null set ◦ P(A∪B)=P(A)+P(B) 4STA 291 Summer 2008 Lecture 7

 Can be difficult  Different approaches to assigning probabilities to events ◦ Subjective ◦ Objective  Equally likely outcomes (classical approach)  Relative frequency 5STA 291 Summer 2008 Lecture 7

6  Relies on a person to make a judgment as to how likely an event will occur ◦ Events of interest are usually events that cannot be replicated easily or cannot be modeled with the equally likely outcomes approach  As such, these values will most likely vary from person to person  The only rule for a subjective probability is that the probability of the event must be a value in the interval [0,1] STA 291 Summer 2008 Lecture 7

 The equally likely approach usually relies on symmetry to assign probabilities to events ◦ As such, previous research or experiments are not needed to determine the probabilities  Suppose that an experiment has only n outcomes  The equally likely approach to probability assigns a probability of 1/n to each of the outcomes  Further, if an event A is made up of m outcomes then P(A) = m/n STA 291 Summer 2008 Lecture 77

 Borrows from calculus’ concept of the limit ◦ We cannot repeat an experiment infinitely many times so instead we use a ‘large’ n  Process  Repeat an experiment n times  Record the number of times an event A occurs, denote this value by a  Calculate the value of a/n 8STA 291 Summer 2008 Lecture 7

 Let A be the event A = {o 1, o 2, …, o k }, where o 1, o 2, …, o k are k different outcomes  Suppose the first digit of a license plate is randomly selected between 0 and 9 ◦ What is the probability that the digit 3? ◦ What is the probability that the digit is less than 4? 9STA 291 Summer 2008 Lecture 7

◦ Note: P(A|B) is read as “the probability that A occurs given that B has occurred” 10STA 291 Summer 2008 Lecture 7

 If events A and B are independent, then the events have no influence on each other ◦ P(A) is unaffected by whether or not B has occurred ◦ Mathematically, if A is independent of B  P(A|B)=P(A)  Multiplication rule for independent events A and B ◦ P(A∩B)=P(A)P(B) 11STA 291 Summer 2008 Lecture 7

 Flip a coin twice, what is the probability of observing two heads?  Flip a coin twice, what is the probability of observing a head then a tail? A tail then a head? One head and one tail?  A 78% free throw shooter is fouled while shooting a three pointer, what is the probability he makes all 3 free throws? None? 12STA 291 Summer 2008 Lecture 7

 X is a random variable if the value that X will assume cannot be predicted with certainty ◦ That’s why its called random  Two types of random variables ◦ Discrete  Can only assume a finite or countably infinite number of different values ◦ Continuous  Can assume all the values in some interval 13STA 291 Summer 2008 Lecture 7

 Are the following random variables discrete or continuous? ◦ X = number of houses sold by a real estate developer per week ◦ X = weight of a child at birth ◦ X = time required to run 800 meters ◦ X = number of heads in ten tosses of a coin 14STA 291 Summer 2008 Lecture 7

 A list of the possible values of a random variable X, say (x i ) and the probability associated with each, P(X=x i ) ◦ All probabilities must be nonnegative ◦ Probabilities sum to 1 15STA 291 Summer 2008 Lecture 7

 The table above gives the proportion of employees who use X number of sick days in a year ◦ An employee is to be selected at random  Let X = # of days of leave  P(X=2) =  P(X≥4) =  P(X<4) =  P(1≤X≤6) = 16 X P(X) STA 291 Summer 2008 Lecture 7

 Expected Value (or mean) of a random variable X ◦ Mean = E(X) = μ = Σx i P(X=x i )  Example ◦ E(X) = 17 X P(X) STA 291 Summer 2008 Lecture 7

 Variance ◦ Var(X) = E(X-μ) 2 = σ 2 = Σ(x i -μ) 2 P(X=x i )  Example ◦ Var(X) = 18 X P(X) STA 291 Summer 2008 Lecture 7

 A random variable X is called a Bernoulli r.v. if X can only take either the value 0 (failure) or 1 (success)  Heads/Tails  Live/Die  Defective/Nondefective ◦ Probabilities are denoted by  P(success) = P(1) = p  P(failure) = P(0) = 1-p = q ◦ Expected value of a Bernoulli r.v. = p ◦ Variance = pq 19STA 291 Summer 2008 Lecture 7

 Suppose we perform several, we’ll say n, Bernoulli experiments and they are all independent of each other (meaning the outcome of one even doesn’t effect the outcome of another) ◦ Label these n Bernoulli random variables in this manner: X 1, X 2,…,X n  The probability of success in a single trial is p  The probability of success doesn’t change from trial to trial  We will build a new random variable X using all of these Bernoulli random variables: ◦ What are the possible outcomes of X? What is X counting? 20STA 291 Summer 2008 Lecture 7

 The probability of observing k successes in n independent trails is ◦ Assuming the probability of success is p ◦ Note:  Why do we need this? 21STA 291 Summer 2008 Lecture 7

 For small n, the Binomial coefficient “n choose k” can be derived without much mathematics 22STA 291 Summer 2008 Lecture 7

 Assume Zolton is a 68% free throw shooter ◦ What is the probability of Zolton making 5 out of 6 free throws? ◦ What is the probability of Zolton making 4 out of 6 free throws? 23STA 291 Summer 2008 Lecture 7

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