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Lecture 2 Probability – some examples – sources of bias – types of probability Problems Expected Values Conditional Probability Important probability formulas An exercise first Answer the questions on the sheet of paper that is handed to you (should take less than 1 minute). You’re not being tested on your general knowledge -- I just want to get your estimate. Excel functions last week Calculate: – mean using =AVERAGE() – median using =MEDIAN() – mode using =MODE() – standard deviation using =STDEV() – variance using =VAR() – Get all these at once using >Tools>Data Analysis> Descriptive statistics Some thought exercises about Probability Birthday Problem What is the chance of two people in this class having the same birthday? Causes of Death Which is more likely: being killed by aeroplane parts falling from the sky or being killed by a shark? Which is more likely: dying from stomach cancer or dying in a car accident? Cancer Screening A doctor has just examined a woman for breast cancer. She has a lump in her breast, but based on years of experience the doctor estimates the odds of malignancy at 1 in 100. A mammogram is ordered. In general these tests accurately classify 80% of malignant tumours and 90% of benign tumours. If the mammogram says the tumour is malignant, what would you say the overall chances of the tumour being malignant are now? Estimating probabilities: sources of bias Representativeness Conservatism Availability Risk perceptions Confusion of the Inverse Anchoring Looking on the Bright Side Coincidences Compound Events For more details see “The Psychology of Judgement and Decision Making” by Plous Report back on homework How closely did the probability estimates agree? Why do you think they differed? How would you go about coming up with a single number? Using pictures to understand probabilities Venn Diagrams – great way to visualise simple probabilities and overlapping events Probability Trees – useful for sequential events, repeated events, multiple items or people Contingency Tables – useful for understanding, visualising and computing conditional probabilities Venn Diagrams Venn diagrams are used to visually convey the concept of sample spaces and events. A rectangle is used to represent the sample space of an experiment and as such contains all possible sample points. Circles within the rectangle represent events. The area within the circle contains the sample points which belong to that event. Simple Venn Diagram A Ac Venn Diagram A B A Bc A B Ac B Ac Bc Venn Diagram of Union of Events A and B AB Ac Bc Share price problem P(A up) = 0.7, P(B up) = 0.5 P(C up) = 0.3, P(A up B up) = 0.3 P(A up C up) = 0.1, P(B up C up) = 0.2 P(A up B up C up) = 0.05 What is P(A up B up) ? What is P(A up B up C up) ? What is P(A up B down C down) ? Probability Trees Each node is an experiment. Each branch is an outcome. Branches that come out of a node should be exhaustive and mutually Conditional exclusive. probabilities Prob= 0.7 Prob= 0.6 YY 0.42 Y 0.6 Prob= 0.3 YN 0.18 1 Prob= 0.4 Prob= 0.4 NY 0.16 N 0.4 Prob= 0.6 NN 0.24 Job offers Joint Probability Table Event A Ac B P(A B) P(Ac B) P(B) Event Bc P(A Bc) P(Ac Bc) P(Bc) P(A) P(Ac) 1 Convert Cross-Tabs Table Into a Joint Probability Table No JS JS Cross-Tabs LOW 13 4 PROD Table (n = 36) HIGH PROD 1 18 Total = 36 No JS JS Joint LOW .361 .111 4/36 PROD Probability Table HIGH PROD .028 .500 Joint and Union Probabilities No JS JS LOW PROD .361 .111 .472 HIGH PROD .028 .500 .528 .389 .611 1.00 Unconditional Probability P(LOW PROD) = .472 Joint Probability P(LOW PROD JS) = .111 Union Probability P(LOW PROD JS) = .361 + .111 + .500 = .972 Excel - Probability Pivot-tables Downloadable spreadsheet from webpage with “proforma” probability tables Scholarships In a study of 100 students who had been awarded university scholarships, it was found that 40 had part-time jobs, 25 had achieved honours in the previous semester, and 15 had both a part-time job and had achieved honours. What was the probability that a student had a part-time job or had achieved honours (ie. the union probability)? Expected Value What is an expected value? Notation: expected value of x is E(x) Calculating E(x): E ( x) P( x1 ) x1 P( x2 ) x2 P( x3 ) x3 ... Expected Value example If you want to rent out a house and you know: Weekly Rent Probability $210 0.4 $225 0.3 $240 0.25 $250 0.05 What do you expect to earn in rent each week? Watch this space (Video) Use probability language to describe the two main positions in this argument Which side do agree with? Why? Can you think of other examples where there are similar differences in interpreting probabilities and expected values? Monty Hall/Let’s Make a Deal Door A Door B Door C Conditional Probability P(A|B) is the probability of A happening given that B has happened. This is called a conditional probability Important formula: (Bayes’ Theorem) P(A|B) = P(A B) / P(B) Or round the other way: P(A B) = P(A|B) x P(B) Unconditional and Conditional Probabilities No JS JS LOW .361 .111 .472 PROD HIGH PROD .028 .500 .528 .389 .611 Unconditional Probability P(HIGH PROD) = .528 Conditional Probability P(HIGH PROD | JS) = .500/.611 = .81 Only Interested in JS Groups. Thus .611 is the Denominator. High Prod is .500, the Numerator. Independent Events Events are independent if the outcome of one does not affect the outcome of the other. Example: – the profit made by Telstra is independent of what I chose to have for breakfast yesterday. – But the profit made by Telstra is not independent of whether I chose to phone my friend in California yesterday. Statistical Independence ? If P(A) = P(A | B), then Events A and B are Independent. – Does P(High Prod) = P(High Prod | Job Switch)? Knowing Event B Happened Does Not Affect the Probability of A. Is Job Switching Related to Productivity Level? P(HIGH PROD) = .528 P(HIGH PROD | JS) = .500/.611 = .818 P(HIGH PROD | NJS) = .028/.389 = .072 Note: Three Probabilities are Different. Probability of High Productivity if Group Members Switch Jobs is .818. Probability of High Productivity if Group Members Do Not Switch Jobs is Only .072. Conclusion: High Productivity and Job Switching May be Related. Probability Trees – independent events Trees can also be used when events are independent. Probabilities on branches are just simple probabilities of each event occurring This is because literal (symbolic) definition of independence is: P(A) = P(A | B) Conditional probability is simple probability for independent events Mutually Exclusive Events Events are mutually exclusive if they both couldn’t happen at once. Example: – Telstra making a profit of more than $300 million is mutually exclusive with them making a profit of less than $300 million this year. – Me having toast for breakfast is not mutually exclusive with me having Weetbix for breakfast (I might have both). Calculating Union probabilities In general the probability of A B happening is: P(A B) = P(A) + P(B) - P(A B) If events are mutually exclusive then the probability of A B happening is: P(A B) = P(A) + P(B) Calculating Intersection probabilities In general the probability of A B happening must be found by gathering data to estimate it separately. However if events are independent then the probability of A and B happening is: P(A B) = P(A) P(B) Murder He Wrote Do you understand the probabilities that are talked about in the article? What does the law mean when it talks about “beyond a reasonable doubt”? Can you assign a probability to this? The Birthday Problem Birthday Problem 1 0.9 0.8 P(at least one match) 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 1 4 7 Number of People (n) Birthday Problem (the calculations) Probability of 2 people not having the same birthday = 364/365 Probability of 3 people not having the same birthday = Probability of 2 people not having the same birthday AND a 3rd person not having the same birthday as either of the first 2 = 364/365 363/365 Probability of 4 people not having the same birthday = 364/365 363/365 362/365 Birthday problem (cont.) Prob of no birthday matches with (n+1) people = 364 363 362 ... (365 n) 365n Prob of (n+1) people having at least one matching birthday = 1 - P(no matches) Example: P(no matches for 10 people) = 0.8831 P(at least one match for 10 people) = 0.1169 Wallet Problem What is the 5 5 probability that given we select a five dollar bill from one of these 5 10 three wallets at random that the other bill in that wallet is a 10 10 five? Probability Trees Remember we used probability trees with conditional probabilities on the branches. Useful to depict sequential events. P(C|A)=0.4 AC 0.08 P(A)=0.2 A 0.2 P(D|A)=0.6 AD 0.12 1 P(C|B)=0.3 BC 0.24 P(B)=0.8 B 0.8 P(D|B)=0.7 BD 0.56 Conditional probabilities Can also think of a conditional probability as a probability of something happening within a “subset” of the population. Use a Venn diagram or a probability table e.g. P(A | B) = 10 / 50 = 0.2 A B 20 10 40 (Notice that this is identical to 30 Bayes theorem formula) Another way of thinking about conditional probabilities Many times during probability analysis, we may want to revise some prior probabilities P(A) based on new information (B). Bayes Theorem allows us to do this. The new probabilities are called posterior probabilities P(A|B) and, as the new estimates of the likelihood of A, can be used in future calculations instead of P(A). Excel – Bayes Theorem Download Bayes Theorem spreadsheet from webpage to do the calculations Important Formulas P ( A) P ( A ) 1 c P ( A B ) P ( A B ) P ( A) c P ( A B ) P ( A) P ( B ) P ( A B ) P ( A B ) P ( B A) P( B) P( A B) P( B) P( A B) Rashes A pharmaceutical company conducted a study to evaluate the effect of an allergy relief medicine; 250 patients with symptoms that included itchy eyes and a skin rash were given a new drug. 90 of the patients experienced eye relief, 135 had their rash clear up, and 45 had relief from both conditions. What is the likelihood that a patient will experience relief from at least one of these conditions? Do these events appear independent? Explain. Rush Orders Rush orders for a raw material are placed with two different suppliers, A and B. If neither order arrives in 4 days production is shut down until at least one comes. P(A will deliver in 4 days) = 0.55, P(B will deliver in 4 days) = 0.35 P(both deliver in 4 days)? P(at least one delivers in 4 days)? P(production is shut down)? Advertising Soap B = bought, S = remembers seeing ad P(B) = 0.2, P(S) = 0.4, P(B S) = 0.12 P(B|S)? Does seeing the ad increase probability of purchasing? Would you keep advertising? If those who don’t buy from you, buy from your competitor: What would be your estimate of your market share? Would your continuing the ad increase your market share? Why or why not? Advertising Soap (cont.) Another ad has been tested and has values of P(S)=0.3 and P(B S) = 0.1. What is P(B|S) for this ad? Which ad has a bigger effect on purchases? Credit? A bank is reviewing its credit card policy with a view to recalling some of its credit cards. In the past approximately 5% of cardholders have defaulted and the bank has been unable to collect the outstanding balance. The bank has further found that the probability of missing one or more payments for those customers who do not default is 0.20. Of course the probability of missing one or more payments for those who default is 1. Given that a customer has missed a payment, compute the probability that they will default. Integrated Circuit chips A process produces IC chips. Over the long-run the fraction of bad chips produced by the process is around 20%. Thorough testing is expensive but there is a cheap test. All good chips pass the cheap test but so do 10% of bad chips. Given that a chip passes the test what is the probability that it is a good chip? If a company sold all chips that passed the test what fraction of what they sold would be bad? Cancer Screening Use Bayes Theorem to answer the cancer screening question from beginning of lecture. Prior probability of malignancy = 0.01 P(positive test | malignant) = 0.8 P(negative test | benign) = 0.9 What is the posterior probability of malignancy? (I.e. the new probability of malignancy if you get information that the test was positive) What did we do? Discussed where intuition about probability fails Discussed how probability is estimated Venn Diagrams Explored the use of tree diagrams and probability tables in calculating probability. Talked about independence and mutual exclusivity Expected value Conditional probability Talked about statistical independence Worked on problems Excel functions today Expected values using =SUMPRODUCT() Bayes theorem and probability tables spreadsheet – download and use to automate simple calculations. Practice using this to solve problems you get in class and in textbook/webpage. Managerial applications What did you learn today that makes a difference to the way you manage? What are the three most important things to remember from today’s lecture? Next class Prepare case study “Who is the customer?” (Specific questions given on the class webpage). This is assessed if your syndicate chose option 1 Read and answer questions for “Hair Tonic” Read supplementary readings on Decision Trees and Valuing Information Answer additional questions on webpage

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