STA 291 Spring 2008 Lecture 7 Dustin Lueker

Probability Terminology 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 STA 291 Spring 2008 Lecture 7

Basic Concepts 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(Ac)=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 STA 291 Spring 2008 Lecture 7

Probability 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) STA 291 Spring 2008 Lecture 7

Assigning Probabilities to Events Can be difficult Different approaches to assigning probabilities to events Objective Equally likely outcomes (classical approach) Relative frequency Subjective STA 291 Spring 2008 Lecture 7

Equally Likely Approach The equally likely outcomes 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 Spring 2008 Lecture 7

Relative Frequency Approach 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 STA 291 Spring 2008 Lecture 7

Subjective Probability Approach 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 Spring 2008 Lecture 7

Probabilities of Events Let A be the event A = {o1, o2, …, ok}, where o1, o2, …, ok 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? STA 291 Spring 2008 Lecture 7

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

Independence 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) STA 291 Spring 2008 Lecture 7

Example 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? STA 291 Spring 2008 Lecture 7

Random Variables 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 STA 291 Spring 2008 Lecture 7

Examples 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 STA 291 Spring 2008 Lecture 7

Discrete Probability Distribution A list of the possible values of a random variable X, say (xi) and the probability associated with each, P(X=xi) All probabilities must be nonnegative Probabilities sum to 1 STA 291 Spring 2008 Lecture 7

Example X 1 2 3 4 5 6 7 P(X) .1 .2 .15 .05 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) = STA 291 Spring 2008 Lecture 7

Expected Value of a Discrete Random Variable Expected Value (or mean) of a random variable X Mean = E(X) = μ = ΣxiP(X=xi) Example E(X) = X 2 4 6 8 10 12 P(X) .1 .05 .4 .25 STA 291 Spring 2008 Lecture 7

Variance of a Discrete Random Variable Var(X) = E(X-μ)2 = σ2 = Σ(xi-μ)2P(X=xi) Example Var(X) = X 2 4 6 8 10 12 P(X) .1 .05 .4 .25 STA 291 Spring 2008 Lecture 7

Bernoulli Random Variables 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 STA 291 Spring 2008 Lecture 7

Binomial Distribution 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: X1, X2,…,Xn 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? STA 291 Spring 2008 Lecture 7

Binomial Distribution The probability of observing k successes in n independent trails is Assuming the probability of success is p Note: Why do we need this? STA 291 Spring 2008 Lecture 7

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

Example 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? STA 291 Spring 2008 Lecture 7

Binomial Distribution Properties STA 291 Spring 2008 Lecture 7