Statistics Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability Where We’ve Been Making Inferences about a Population Based on a Sample Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability Where We’re Going Probability as a Measure of Uncertainty Basic Rules for Finding Probabilities Probability as a Measure of Reliability for an Inference McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability A sample point is the most basic outcome of an experiment. An Ace A four A Head McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability A sample space of an experiment is the collection of all sample points. Roll a single die: S: {1, 2, 3, 4, 5, 6} McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability Sample points and sample spaces are often represented with Venn diagrams. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability Probability Rules for Sample Points All probabilities must be between 0 and 1. The probabilities of all the sample points must sum to 1. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability Probability Rules for Sample Points All probabilities must be between 0 and 1 The probabilities of all the sample points must sum to 1 0 indicates an impossible outcome and 1 a certain outcome. Something has to happen. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability An event is a specific collection of sample points: Event A: Observe an even number. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability The probability of an event is the sum of the probabilities of the sample points in the sample space for the event. Event A: Observe an even number. P(A) = 1/6 + 1/6 + 1/6 = 3/6 = 1/2 McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability Calculating Probabilities for Events Define the experiment. List the sample points. Assign probabilities to sample points. Collect all sample points in the event of interest. The sum of the sample point probabilities is the event probability. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability Calculating Probabilities for Events Define the experiment List the sample points Assign probabilities to sample points Collect all sample points in the event of interest The sum of the sample point probabilities is the event probability If the number of sample points gets too large, we need a way to keep track of how they can be combined for different events. McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability If a sample of n elements is drawn from a set of N elements (N ≥ n), the number of different samples is McClave: Statistics, 11th ed. Chapter 3: Probability

3.1: Events, Sample Spaces and Probability If a sample of 5 elements is drawn from a set of 20 elements, the number of different samples is McClave: Statistics, 11th ed. Chapter 3: Probability

3.2: Unions and Intersections McClave: Statistics, 11th ed. Chapter 3: Probability

3.2: Unions and Intersections McClave: Statistics, 11th ed. Chapter 3: Probability

3.2: Unions and Intersections A  C A  B  C B C McClave: Statistics, 11th ed. Chapter 3: Probability

3.3: Complementary Events The complement of any event A is the event that A does not occur, AC. A: {Toss an even number} AC: {Toss an odd number} B: {Toss a number ≤ 3} BC: {Toss a number ≥ 4} A  B = {1,2,3,4,6} [A  B]C = {5} (Neither A nor B occur) McClave: Statistics, 11th ed. Chapter 3: Probability

3.3: Complementary Events McClave: Statistics, 11th ed. Chapter 3: Probability

3.3: Complementary Events A: {At least one head on two coin flips} AC: {No heads} McClave: Statistics, 11th ed. Chapter 3: Probability

3.4: The Additive Rule and Mutually Exclusive Events The probability of the union of events A and B is the sum of the probabilities of A and B minus the probability of the intersection of A and B: McClave: Statistics, 11th ed. Chapter 3: Probability

3.4: The Additive Rule and Mutually Exclusive Events At a particular hospital, the probability of a patient having surgery (Event A) is .12, of an obstetric treatment (Event B) .16, and of both .02. What is the probability that a patient will have either treatment? McClave: Statistics, 11th ed. Chapter 3: Probability

3.4: The Additive Rule and Mutually Exclusive Events Events A and B are mutually exclusive if A  B contains no sample points. McClave: Statistics, 11th ed. Chapter 3: Probability

3.4: The Additive Rule and Mutually Exclusive Events If A and B are mutually exclusive, McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability Additional information or other events occurring may have an impact on the probability of an event. McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability Additional information may have an impact on the probability of an event. P(Rolling a 6) is one-sixth (unconditionally). If we know an even number was rolled, the probability of a 6 goes up to one-third. McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability The sample space is reduced to only the conditioning event. To find P(A), once we know B has occurred (i.e., given B), we ignore BC (including the A region within BC). B BC A A McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability BC A A McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability 55% of sampled executives had cheated at golf (event A). P(A) = .55 20% of sampled executives had cheated at golf and lied in business (event B). P(A  B) = .20 What is the probability that an executive had lied in business, given s/he had cheated in golf, P(B|A)? McClave: Statistics, 11th ed. Chapter 3: Probability

3.5: Conditional Probability P(A) = .55 P(A  B) = .20 What is P(B|A)? McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events The conditional probability formula can be rearranged into the Multiplicative Rule of Probability to find joint probability. McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events Assume three of ten workers give illegal deductions Event A: {First worker selected gives an illegal deduction} Event B: {Second worker selected gives an illegal deduction} P(A) = P(B) = .1 + .1 + .1 = .3 P(B|A) has only nine sample points, and two targeted workers, since we selected one of the targeted workers in the first round: P(B|A) = 1/9 + 1/9 = .11 + .11 = .22 The probability that both of the first two workers selected will have given illegal deductions P(AB) = P(B|A)P(A) = .(3) (.22) = .066 McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events 10 Workers Illegal Deductions P = .3 P = .3 x.22 = .066 No Illegal Deductions P = .3 x.78 = .234 No Illegal P = .7 P = .7 x.33 = .231 P = .7 x .67 = .469 A Tree Diagram First selected worker Second selected worker McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Independent Events P(A|B) = P(A) P(B|A) = P(B)  Studying stats 40 hours per week Working 40 hours per week Having blue eyes McClave: Statistics, 11th ed. Chapter 3: Probability

3.6: The Multiplicative Rule and Independent Events P(A|B) ≠ P(A) P(B|A) ≠ P(B) Independent Events P(A|B) = P(A) P(B|A) = P(B) Mutually exclusive events are dependent: P(B|A) = 0 Since P(B|A) = P(B), P(AB) = P(A)P(B|A) = P(A)P(B) McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability 3.7: Random Sampling If n elements are selected from a population in such a way that every set of n elements in the population has an equal probability of being selected, the n elements are said to be a (simple) random sample. McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability 3.7: Random Sampling How many five-card poker hands can be dealt from a standard 52-card deck? Use the combination rule: McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability 3.7: Random Sampling Random samples can be generated by Mixing up the elements and drawing by hand, say, out of a hat (for small populations) Random number generators Random number tables Table I, App. B Random sample/number commands on software McClave: Statistics, 11th ed. Chapter 3: Probability

3.8: Some Additional Counting Rules Situation Number of Different Results Multiplicative Rule Draw one element from each of k sets, sized n1, n2, n3, … nk Permutations Rule Draw n elements, arranged in a distinct order, from a set of N elements Partitions Rule Partition N elements into k groups, sized n1, n2, n3, … nk (ni=N) McClave: Statistics, 11th ed. Chapter 3: Probability

3.8: Some Additional Counting Rules Multiplicative Rule Assume one professor from each of the six departments in a division will be selected for a special committee. The various departments have four, seven, six, eight, six and five professors eligible. How many different committees, X*, could be formed? McClave: Statistics, 11th ed. Chapter 3: Probability

3.8: Some Additional Counting Rules Permutations If there are eight horses entered in a race, how many different win, place and show possibilities are there? McClave: Statistics, 11th ed. Chapter 3: Probability

3.8: Some Additional Counting Rules Partitions Rule The technical director of a theatre has twenty stagehands. She needs eight electricians, ten carpenters and two props people. How many different allocations of stagehands, Y*, can there be? McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability 3.8: Bayes’s Rule Given k mutually exclusive and exhaustive events B1, B2,… Bk, and an observed event A, then McClave: Statistics, 11th ed. Chapter 3: Probability

McClave: Statistics, 11th ed. Chapter 3: Probability 3.8: Bayes’s Rule Suppose the events B1, B2, and B3, are mutually exclusive and complementary events with P(B1) = .2, P(B2) = .15 and P(B3) = .65. Another event A has these conditional probabilities: P(A|B1) = .4, P(A|B2) = .25 and P(A|B3) = .6. What is P(B1|A)? McClave: Statistics, 11th ed. Chapter 3: Probability