Introductory Statistics. Lecture 13

The Multiplication Rule Math 140 Introductory Statistics „ The probability that event A and event B both happen is given by P (A and B) = P (A).P (B | A) or alternatively P (A and B) = P (B).P (A | B) Lecture 13 Sections 5.4,5.5 Definition of Conditional Probability Important Uses of Conditional Probability „ For any two events A and B, P (B)>0, P (A and B ) P (B ) „ Example: Suppose you roll two dice. Use the definition of conditional probability to find the probability that you get a sum of 8 given that you rolled doubles. P (A | B ) = P (sum 8 and doubles ) P (sum 8 | doubles ) = P (doubles) 1 / 36 1 = = 6 / 36 6 „ To compare sampling with or without replacement. „ To study effectiveness of medical tests „ To study effectiveness of statistical testing 1 Conditional Probability and Medical Tests Conditional Probability and Medical Tests Test Result „ In medicine, screening tests give a quick indication of whether or not a person is likely to have a particular disease. Because screening tests are intended to be relatively quick and noninvasive, they are often not as accurate as other tests that take longer or are more invasive. „ A two-way table is often used to show the four possible outcomes of a screening test. Test Result Disease Positive Negative Total Present a b a+b Absent c d c+d Total a+c b+d a+b+c+d Test Result Total Present a b a+b Absent c d c+d Total a+c b+d a+b+c+d P (no disease | test positive) = „ False Negative Rate = c a+c P (disease present | test negative) = „ Sensitivity = P (test positive | disease present) = „ Specificity = P (test negative | no disease) = a a+b b b +d d c +d Example P34 (p. 335) „ A laboratory technician is being tested on her ability to detect Positive Negative Total Present 9 1 10 Absent 50 9,940 9,990 Total 59 9,941 10,000 50 ≈ 0.8474 59 1 ≈ 0.0001 „ False Neg. Rate = P (disease present | test negative) = 9941 9 = 0.90 „ Sensitivity = P (test positive | disease present) = 10 9940 „ Specificity = P (test negative | no disease) = ≈ .9949 9990 „ False Pos. Rate = Negative „ False Positive Rate = Example of a Rare Disease on 10,000 patients Disease Disease Positive contaminated blood samples. Among 100 samples given to her, 20 are contaminated, each with about the same degree of contamination. Suppose the technician makes the correct decision 90% of the time (regardless of contamination or not). Make a table showing what you would expect to happen. What is her false positive rate? What is her false negative rate? How would these rates change if she were given 100 samples with 50 contaminated? Detection of Contamination (test) P (no disease | test positive) = Positive Contaminated? Negative Total Yes 18 2 20 No 8 72 80 Total 26 74 100 2 Conditional Probability and Statistical Inference Example P34 (p. 335) Detection of Contamination (test) Positive Contaminated? Negative „ Statistician: Suppose you draw Total Yes 18 2 20 No 8 72 80 Total 26 74 100 „ „ „ False Pos. Rate = 8/26 = 0.3076 „ „ False Neg. Rate = 2/74 = 0.027 „ „ „ Sensitivity = 18/20 = 0.90 „ 3 workers at random from the set of 10 hourly workers. This establishes random sampling as the model for the study. Lawyer: Okay. Statistician: It turns out that there are 120, possible samples of size 3, and only 6 of them give an average age of 58 or more. Lawyer: So the probability is 6/120 , or .05. Statistician: Right. Lawyer: There’s only a 5% chance the company didn’t discriminate and a 95% chance that they did. Statistician: No, that’s not true. „ Specificity = 72/80 = 0.9 5.5 Independent Events does not affect the probability of A happening, and conversely knowing that A happened does not affect the probability of B. P (av. age ≥ 58 | random draws ) = .05 Example: Water, Gender, and Independence (p.340) Identified Tap Water ? Drinks Bottled Water ? Identified Tap Water ? Yes No Total Yes No Total Yes No 24 36 Total 60 „ In other words, knowing that B happened age-neutral model of random draws is correct, then there’s only a 5% chance of getting an average age of 58 or more. „ Lawyer: So the chance the company is guilty must be 95%. „ Statistician: Slow down. If you start by assuming the model is true, you can compute the chances of various results. But you’re trying to start from the results and compute the chance that the model is right or wrong. You can’t do that. P (no discr. | av. age = 58) ?? „ Events A and B are independent if and only if P (A | B)= P (A). Equivalently, A and B are independent if and only if P (B | A)= P (B). „ Lawyer: But you said . . . „ Statistician: I said that if the 6 30 34 70 40 100 Male Gender 21 14 35 Female 39 26 65 Total 40 100 60 „ Show that the events is a male and correctly identifies tap water are independent and that the events drinks bottled water and correctly identifies tap water are not independent. 3