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3.1 Background(p88) In everyday conversation, what does the term “probability” measure? I do not play Lotto … the chances to win is too small Mr president, the chances that sales will decrease if we increase prices are high What is the chance that the new investment will be profitable? How likely is it that the project will be finished on time? 3.1 Background(p88) “In everyday conversation, the term probability is a measure of one’s belief in the occurrence of a future event” NEW YORK, Mon: Mr. Webster Todd, Chairman of the American National Transportation Safety Board, said today that the chances of two jumbo jets colliding on the ground were about 6 million to one... –AAP Professor Speed, who had strong research interests in probability, was intrigued by this statement and wondered how the board had calculated their figure. Speed wrote to the chairman. In the reply it was stated that the figure (6 million to one) has no statistical validity nor was it intended to be a rigorous probability statement. 3.1 Background(p88) “…. Six million to one” This is just a numerical measure of the very small likelihood of the event to occur … called a probability. Probability is a numerical measure of the likelihood that an event will occur Impossible 0 Equal likely Certain 1 3.1 Background (p88) Deterministic versus Random experiment 3.1 Background(p88) Statistics Inferential Statistics Descriptive Statistics Inference Probability Theory Randomness 3.1 Background(p88) Experiment: Outcomes: Toss a coin Select a part for inspection Conduct a sales call Roll a die Play a football game Head, Tail Defective, Non-defective Purchase, No-purchase 1, 2, 3, 4, 5, 6 Win, Lose, tie Definition: event An event is an outcome or a set of outcomes of a random experiment Event (Capital letter) = { outcomes described by event} e.g. the event “even number” when a die is rolled: E = { 2, 4, 6 } Probability is a numerical measurement of the likelihood that an event will occur and is denoted as P(outcome) 3.1 Background(p88) Experiment: Outcomes: Toss a coin Select a part for inspection Conduct a sales call Roll a die Play a football game Head, Tail Defective, Non-defective Purchase, No-purchase 1, 2, 3, 4, 5, 6 Win, Lose, tie Definition: sample space(p89) The sample space is the set of ALL possible outcomes: S e.g. S = {1, 2, 3, 4, 5, 6} 3.2 First Principles(p90) Experiment: Roll a die E1: Observe a 1 E2: Observe a 2 E3: Observe a 3 E4: Observe a 4 E5: Observe a 5 E6: Observe a 6 Simple events: They cannot be decomposed - can have one and only one sample point Each outcome is equally likely P(1) = P(2) = … P(6) = 1/6 P(outcome) = 1/N N = total number of outcomes of the experiment A= Observe an odd number P(event) = A {E1, E3 , E5} {1,3,5} (# outcomes in the event) N P(A) = 3/6 3.2 First Principles(p90) 3.2 First Principles(p90) 3.2 First Principles(p90) P( ) Number of “pink” plants Total number of plants = 4/12 = 1/3 3.2 First Principles(p90): Example 3.1 (p91) Subject number 1 2 3 4 5 6 Gender Age M F M F M F 40 42 51 58 67 70 Event P(A) = 3/6 A = Female subjects B = Male subjects C = subjects over the age of 65 P(B) = 3/6 P(C) = 2/6 3.2 First Principles: Some rules and concepts(p91 – p95) Complement rule The union of two events The intersection of two events The additional rule Mutually exclusive The conditional probability rule Independent events 3.2 First Principles: Some rules and concepts(p91 – p95) A graphic technique for visualizing set theory concepts using overlapping circles and shading to indicate intersection, union and complement. It was introduced in the late 1800s by English logician, John Venn, although it is believed that the method originated earlier. 3.2 First Principles: Some rules and concepts(p91 – p95) Set: “B” Is an insect Hatches from an egg Compound eyes Six legs Two pairs of wings Wings straight above when at rest Thin hairless body Have a knob at the end of the antennae Elements of set B Is an insect Hatches from an egg Compound eyes Six legs Two pairs of wings Wings like a tent or flat when at rest Wide furry body Antennae are thick and furry Elements of set M Set: “M” •Is an insect •Hatches from an egg •Compound eyes •Six legs •Two pairs of wings •Wings straight above when at rest •Thin hairless body •Have a knob at the end of the antennae •Is an insect •Hatches from an egg •Compound eyes •Six legs •Two pairs of wings •Wings like a tent or flat when at rest •Wide furry body •Antennae are thick and furry 3.2 First Principles: Some rules and concepts(p91 – p95) •Wings straight above when at rest •Thin hairless body •Have a knob at the end of the antennae •Is an insect •Hatches from an egg •Compound eyes •Six legs •Two pairs of wings •Wings like a tent or flat when at rest •Wide furry body •Antennae are thick and furry 3.2 First Principles: Some rules(p91) Subject number 1 2 3 4 5 6 Gender Age M F M F M F 40 42 51 58 67 70 Event A = Female subjects B = Male subjects C = subjects over the age of 65 C A B 1 2 4 6 5 3 S 3.2 First Principles: Some rules(p91) The Complement of an event: = all outcomes in the sample space that are not in the event P( A) P( A ) 1 P( A) A S C A 2 4 6 5 B 1 3 S P (C ) 4/6 3.2 First Principles: Some rules(p91) The union of A and B is the event containing all sample points belonging to A or B or both. “At least one occurs” A or B The intersection of A and B is the event containing the sample points belonging to both A and B. “AND” “Both events occur” 3.2 First Principles: Some rules(p91) C A 2 4 6 5 B 1 3 S P(Female or over the age of 65) = P( A C ) 4/6 P(Female and over the age of 65) = P( A C ) 1/6 P ( A B) ? A B P ( ) 0 Two events are said to be mutually have no outcomes in common exclusive if they 3.2 First Principles: Some rules(p91) The additional rule C A 2 4 6 5 B 1 3 S P( A B) P( A) P( B) P( A C ) ? P( A C ) P( A) P(C ) P( A C ) 3.2 First Principles: Some rules(p91) Conditional probability The probability of rain today (mid February) is 0.6 It has been raining the whole week. The probability of rain today (mid February) ????? 3.2 First Principles: Some rules(p91) Conditional probability P(“1”) = 1/6 If we know that an odd number has fallen … P(“1”) = 1/3 Conditional Probability 3.2 First Principles: Some rules(p91) 3.2 First Principles: Contingency Table or cross tabulation(p93) 80 70 50 70 30 150 List all possible outcome of the one event List all possible outcome of the other event The sample space 3.2 First Principles: Contingency Table or cross tabulation(p93) A two-way frequency distribution of 220 persons employed by a specific research institution, classified according to type of post and gender is given in the table below: Calculate the probability that a randomly chosen employee: a. Is male P(M)=96/220 b. Is a female researcher P(F R ) 80 220 c. is a female, given that the employee has a management post 3.2 First Principles: Contingency Table or cross tabulation(p93) Educational level of patients seeking care at an allergy clinic Gender Male Female Total Educational Level (years) 0-8 9 - 12 13 - 16 17 + 15 20 17 26 30 42 31 27 45 62 48 53 Total 78 130 208 Suppose a patient is selected at random, what is the probability that the patient Is male? 78/208 Has 9 – 12 years of education? 62/208 Is female and has 9 – 12 years of education? Has at most 12 years of education ? 42/208 P(0-8 or 9-12)=107/208 Is female if we know that the person only have between 9 – 12 years of education? 42/62 3.2 First Principles: Some rules: Conditional probability (p94) P( A B) P( A B) P( B) Educational level of patients seeking care at an allergy clinic Gender Male Female Total Educational Level (years) 0-8 9 - 12 13 - 16 17 + 15 20 17 26 30 42 31 27 45 62 48 53 Total 78 130 208 Is female if we know that the person only have between 9 – 12 years of education? 42/62 P( Female 9 12) P( Female 9 12) P(9 12) 42 42 208 62 62 208 3.2 First Principles: Independence (p95) Two events are said to be independent if the occurrence of one event does not influence the probability of the other P( A B) P( A) P( B A) P( B) P( A B) P( A) P( B) 3.2 First Principles: Independent: Example 3.2B(p95) Educational level of patients seeking care at an allergy clinic Gender Male Female Total Educational Level (years) 0-8 9 - 12 13 - 16 17 + 25 30 25 25 25 30 25 25 50 60 50 50 Total 105 105 210 Are the two events “Male” and “17+” independent? 25 P( Male 17 ) 210 105 50 25 P( Male) P(17 ) 210 210 210 Self study: Example 3.3 3.3 Combinations and permutations (p98) 1 2 3 Sampling With replacement 5 4 6 Sampling Without replacement Order important Order not important Order important Order not important 1 1 1 1 1 1 1 1 1 2 1 2 1 2 1 2 2 1 2 1 3.3 Combinations and permutations (p98) Sampling With replacement Order important Order not important Sampling Without replacement Order important Order not important 3.3 Combinations and permutations (p98) ABC ACB BCA BAC CAB CBA 1 2 3 3 1 2 ABC 3.3 Permutations (p99) When sampling WITHOUT replacement, the number of distinct arrangements (i.e., order important), called permutations of n individuals from a population of N, is given by N! N Pn ( N n)! N! = N(N-1)(N-2) … (3)(2)(1)0! 0! = 1 3.3 Combinations (p100) When sampling WITHOUT replacement, the number of samples in which order is not important, or combinations, of n individuals from a population of size N is given by N! N Cn n!( N n)! 3.3 Permutations and Combinations (p99) 3.3 Random Variable (p100) Definition of a Random variable: A random variable is a numerical description of the outcome of an experiment Experiment Outcomes Numerical Description = Random variable 3.3 Probability Distribution(p102) 1 3 2 4 Sampling Without replacement 1 0 1 1 2 1 0 1 1 2 0 1 1 2 5 6 Order not important Let X = number of females X 0 1 1 2 P(X) 3/15 9/15 3/15 1 The values of the random variable and the corresponding probabilities constitutes a probability distribution 3.3 Probability Distribution(p102) The probability distribution for a random variable describes how probabilities are distributed over the values of the random variable. Consider the experiment of tossing a coin twice and noting the outcome after every toss. Let X = the number of heads The probability distribution of X: 1 2 X H 0 H T 1 2 T H T T H P(X) 1/4 2/4 1/4 1 3.3 Probability Distribution(p102) The probability distribution for a discrete variable Y can be represented by a table or a graph or a formula. The probability distribution of X: H T H T H H T T X P(X) 0 1 2 1/4 2/4 1/4 1 P(x) 0.6 2 P ( X x) (0.5) x (0.5) 2 x x 0.5 0.4 0.3 0.2 0.1 0 0 1 2 for x 0 ,1,2 3.3 Probability Distribution(p102) A psychologist determined that the number of sessions required to obtain the trust of a new patient is either 1, 2 or 3. Let X be a random variable indicating the number of sessions required to gain the patient’s trust. The following probability function has been proposed: x P( X x) for x 1,2, or 3 6 a. Is this probability distribution valid? Explain. b. What is the probability it takes exactly two sessions to gain the patient’s trust? c. What is the probability it takes at least two sessions to gain the patient’s trust?