Statistics Chapter 3 Probability Where Weve Been n

Statistics Chapter 3: Probability

Where We’ve Been n n Making Inferences about a Population Based on a Sample Graphical and Numerical Descriptive Measures for Qualitative and Quantitative Data Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 2

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

3. 1: Events, Sample Spaces and Probability n An experiment is an act or process of observation that leads to a single outcome that cannot be predicted with certainty. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 4

3. 1: Events, Sample Spaces and Probability n A sample point is the most basic outcome of an experiment. An Ace A four A Head Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 5

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

3. 1: Events, Sample Spaces and Probability n Sample points and sample spaces are often represented with Venn diagrams. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 7

3. 1: Events, Sample Spaces and Probability Rules for Sample Points n ¡ All probabilities must be between 0 and 1. ¡ The probabilities of all the sample points must sum to 1. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 8

3. 1: Events, Sample Spaces and Probability Rules for Sample Points n ¡ All probabilities must be between 0 and 1 0 indicates an impossible outcome and 1 a certain outcome. ¡ The probabilities of all the sample points must sum to 1 Something has to happen. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 9

3. 1: Events, Sample Spaces and Probability n An event is a specific collection of sample points: ¡ Event A: Observe an even number. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 10

3. 1: Events, Sample Spaces and Probability n 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 = 3/6 = 1/2 Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 11

3. 1: Events, Sample Spaces and Probability n 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. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 12

3. 1: Events, Sample Spaces and Probability n Calculating Probabilities for Events Define the experiment If¡ the number of sample points List the sample points gets too large, we need a way to ¡ Assign probabilities to sample points keep track of how they can be ¡ Collect all sample points in the event of combined interest for different events. ¡ The sum of the sample point probabilities is the event probability ¡ Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 13

3. 1: Events, Sample Spaces and Probability n If a sample of n elements is drawn from a set of N elements (N ≥ n), the number of different samples is Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 14

3. 1: Events, Sample Spaces and Probability n If a sample of 5 elements is drawn from a set of 20 elements, the number of different samples is Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 15

3. 2: Unions and Intersections Compound Events Made of two or more other events Union A B Either A or B, or both, occur Intersection A B Both A and B occur Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 16

3. 2: Unions and Intersections Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 17

3. 2: Unions and Intersections A A B A C A B C B B C C Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 18

3. 3: Complementary Events n 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) Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 19

3. 3: Complementary Events Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 20

3. 3: Complementary Events A: {At least one head on two coin flips} AC: {No heads} Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 21

3. 4: The Additive Rule and Mutually Exclusive Events n 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: Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 22

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? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 23

3. 4: The Additive Rule and Mutually Exclusive Events n Events A and B are mutually exclusive if A B contains no sample points. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 24

3. 4: The Additive Rule and Mutually Exclusive Events If A and B are mutually exclusive, Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 25

3. 5: Conditional Probability n Additional information or other events occurring may have an impact on the probability of an event. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 26

3. 5: Conditional Probability n 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. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 27

3. 5: Conditional Probability n n 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 Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 28

3. 5: Conditional Probability B BC A A Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 29

3. 5: Conditional Probability B A Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 30

3. 5: Conditional Probability n 55% of sampled executives had cheated at golf (event A). ¡ n 20% of sampled executives had cheated at golf and lied in business (event B). ¡ n P(A) =. 55 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)? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 31

3. 5: Conditional Probability n P(A) =. 55 n P(A B) =. 20 n What is P(B|A)? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 32

3. 6: The Multiplicative Rule and Independent Events n The conditional probability formula can be rearranged into the Multiplicative Rule of Probability to find joint probability. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 33

3. 6: The Multiplicative Rule and Independent Events n n 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} ¡ n P(B|A) has only nine sample points, and two targeted workers, since we selected one of the targeted workers in the first round: ¡ n P(A) = P(B) =. 1 +. 1 =. 3 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 Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 34

3. 6: The Multiplicative Rule and Independent Events Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 35

3. 6: The Multiplicative Rule and Independent Events A Tree Diagram First selected worker Second selected worker 10 Workers Illegal Deductions P =. 3 x. 22 =. 066 No Illegal Deductions P =. 3 x. 78 =. 234 No Illegal Deductions P =. 7 x. 33 =. 231 Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability No Illegal Deductions P =. 7 x. 67 =. 469 36

3. 6: The Multiplicative Rule and Independent Events n Dependent Events ¡ ¡ n P(A|B) ≠ P(A) P(B|A) ≠ P(B) Independent Events ¡ ¡ Studying stats 40 hours per week Working 40 hours per week Having blue eyes P(A|B) = P(A) P(B|A) = P(B) Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 37

3. 7: Random Sampling n 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. Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 39

3. 7: Random Sampling n n How many five-card poker hands can be dealt from a standard 52 -card deck? Use the combination rule: Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 40

3. 7: Random Sampling n 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 n ¡ Table I, App. B Random sample/number commands on software Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 41

3. 8: Some Additional Counting Rules Situation Number of Different Results Multiplicative Rule ¡ Draw one element from each of k sets, sized n 1, n 2, n 3, … 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 n 1, n 2, n 3, … nk ( ni=N) Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 42

3. 8: Some Additional Counting Rules n 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? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 43

3. 8: Some Additional Counting Rules n Permutations If there are eight horses entered in a race, how many different win, place and show possibilities are there? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 44

3. 8: Some Additional Counting Rules n 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? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 45

3. 8: Bayes’s Rule n Given k mutually exclusive and exhaustive events B 1, B 2, … Bk, and an observed event A, then Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 46

3. 8: Bayes’s Rule n n Suppose the events B 1, B 2, and B 3, are mutually exclusive and complementary events with P(B 1) =. 2, P(B 2) =. 15 and P(B 3) =. 65. Another event A has these conditional probabilities: P(A|B 1) =. 4, P(A|B 2) =. 25 and P(A|B 3) =. 6. What is P(B 1|A)? Mc. Clave: Statistics, 11 th ed. Chapter 3: Probability 47