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STA 291 Summer 2010

STA 291 Summer 2010. 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

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STA 291 Summer 2010

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  1. STA 291Summer 2010 Lecture 7 Dustin Lueker

  2. 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 Summer 2010 Lecture 7

  3. 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 Summer 2010 Lecture 7

  4. 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 Summer 2010 Lecture 7

  5. Assigning Probabilities to Events • Can be difficult • Different approaches to assigning probabilities to events • Subjective • Objective • Equally likely outcomes (classical approach) • Relative frequency STA 291 Summer 2010 Lecture 7

  6. 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 Summer 2010 Lecture 7

  7. Equally Likely Approach • 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 2010 Lecture 7

  8. 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 Summer 2010 Lecture 7

  9. 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 Summer 2010 Lecture 7

  10. Conditional Probability • Note: P(A|B) is read as “the probability that A occurs given that B has occurred” STA 291 Summer 2010 Lecture 7

  11. 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 Summer 2010 Lecture 7

  12. 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 Summer 2010 Lecture 7

  13. 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 Summer 2010 Lecture 7

  14. 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 event doesn’t effect the outcome of another) • Label these nBernoulli 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 Summer 2010 Lecture 7

  15. 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 Summer 2010 Lecture 7

  16. Binomial Coefficient • For small n, the Binomial coefficient “n choose k” can be derived without much mathematics STA 291 Summer 2010 Lecture 7

  17. 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 Summer 2010 Lecture 7

  18. Center and Spread of the Binomial Distribution STA 291 Summer 2010 Lecture 7

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