#### Transcript Chapter 5

Section 5.3 Conditional Probability: What’s the Probability of A, Given B? Agresti/Franklin Statistics, 1 of 87 Conditional Probability For events A and B, the conditional probability of event A, given that event B has occurred is: P( A andB) P (A | B) P( B) Agresti/Franklin Statistics, 2 of 87 Conditional Probability Agresti/Franklin Statistics, 3 of 87 Example: What are the Chances of a Taxpayer being Audited? Agresti/Franklin Statistics, 4 of 87 Example: Probabilities of a Taxpayer Being Audited Agresti/Franklin Statistics, 5 of 87 Example: Probabilities of a Taxpayer Being Audited What was the probability of being audited, given that the income was ≥ $100,000? • Event A: • Event B: Taxpayer is audited Taxpayer’s income ≥ $100,000 Agresti/Franklin Statistics, 6 of 87 Example: Probabilities of a Taxpayer Being Audited P(A and B) 0.0010 P(A| B) 0.007 P(B) 0.1334 Agresti/Franklin Statistics, 7 of 87 Example: The Triple Blood Test for Down Syndrome A positive test result states that the condition is present A negative test result states that the condition is not present Agresti/Franklin Statistics, 8 of 87 Example: The Triple Blood Test for Down Syndrome False Positive: Test states the condition is present, but it is actually absent False Negative: Test states the condition is absent, but it is actually present Agresti/Franklin Statistics, 9 of 87 Example: The Triple Blood Test for Down Syndrome A study of 5282 women aged 35 or over analyzed the Triple Blood Test to test its accuracy Agresti/Franklin Statistics, 10 of 87 Example: The Triple Blood Test for Down Syndrome Agresti/Franklin Statistics, 11 of 87 Example: The Triple Blood Test for Down Syndrome Assuming the sample is representative of the population, find the estimated probability of a positive test for a randomly chosen pregnant woman 35 years or older Agresti/Franklin Statistics, 12 of 87 Example: The Triple Blood Test for Down Syndrome P(POS) = 1355/5282 = 0.257 Agresti/Franklin Statistics, 13 of 87 Example: The Triple Blood Test for Down Syndrome Given that the diagnostic test result is positive, find the estimated probability that Down syndrome truly is present Agresti/Franklin Statistics, 14 of 87 Example: The Triple Blood Test for Down Syndrome P(D and POS) 48 / 5282 P(D| POS) P(POS) 1355/ 5282 0.009 0.035 0.257 Agresti/Franklin Statistics, 15 of 87 Example: The Triple Blood Test for Down Syndrome Summary: Of the women who tested positive, fewer than 4% actually had fetuses with Down syndrome Agresti/Franklin Statistics, 16 of 87 Multiplication Rule for Finding P(A and B) For events A and B, the probability that A and B both occur equals: • P(A and B) = P(A|B) x P(B) • also P(A and B) = P(B|A) x P(A) Agresti/Franklin Statistics, 17 of 87 Example: How Likely is a Double Fault in Tennis? Roger Federer – 2004 men’s champion in the Wimbledon tennis tournament • He made 64% of his first serves • He faulted on the first serve 36% of the • time Given that he made a fault with his first serve, he made a fault on his second serve only 6% of the time Agresti/Franklin Statistics, 18 of 87 Example: How Likely is a Double Fault in Tennis? Assuming these are typical of his serving performance, when he serves, what is the probability that he makes a double fault? Agresti/Franklin Statistics, 19 of 87 Example: How Likely is a Double Fault in Tennis? P(F1) = 0.36 P(F2|F1) = 0.06 P(F1 and F2) = P(F2|F1) x P(F1) = 0.06 x 0.36 = 0.02 Agresti/Franklin Statistics, 20 of 87 Sampling Without Replacement Once subjects are selected from a population, they are not eligible to be selected again Agresti/Franklin Statistics, 21 of 87 Example: How Likely Are You to Win the Lotto? In Georgia’s Lotto, 6 numbers are randomly sampled without replacement from the integers 1 to 49 You buy a Lotto ticket. What is the probability that it is the winning ticket? Agresti/Franklin Statistics, 22 of 87 Example: How Likely Are You to Win the Lotto? P(have all 6 numbers) = P(have 1st and 2nd and 3rd and 4th and 5th and 6th) = P(have 1st)xP(have 2nd|have 1st)xP(have 3rd| have 1st and 2nd) …P(have 6th|have 1st, 2nd, 3rd, 4th, 5th) Agresti/Franklin Statistics, 23 of 87 Example: How Likely Are You to Win the Lotto? 6/49 x 5/48 x 4/47 x 3/46 x 2/45 x 1/44 = 0.00000007 Agresti/Franklin Statistics, 24 of 87 Independent Events Defined Using Conditional Probabilities Two events A and B are independent if the probability that one occurs is not affected by whether or not the other event occurs Agresti/Franklin Statistics, 25 of 87 Independent Events Defined Using Conditional Probabilities Events A and B are independent if: P(A|B) = P(A) If this holds, then also P(B|A) = P(B) Also, P(A and B) = P(A) x P(B) Agresti/Franklin Statistics, 26 of 87 Checking for Independence Here are three ways to check whether events A and B are independent: • Is P(A|B) = P(A)? • Is P(B|A) = P(B)? • Is P(A and B) = P(A) x P(B)? If any of these is true, the others are also true and the events A and B are independent Agresti/Franklin Statistics, 27 of 87 Example: How to Check Whether Two Events are Independent The diagnostic blood test for Down syndrome: POS = positive result NEG = negative result D = Down Syndrome DC = Unaffected Agresti/Franklin Statistics, 28 of 87 Example: How to Check Whether Two Events are Independent Blood Test: Status POS NEG Total D 0.009 0.001 0.010 Dc 0.247 0.742 0.990 Total 0.257 0.743 1.000 Agresti/Franklin Statistics, 29 of 87 Example: How to Check Whether Two Events are Independent Are the events POS and D independent or dependent? • Is P(POS|D) = P(POS)? Agresti/Franklin Statistics, 30 of 87 Example: How to Check Whether Two Events are Independent Is P(POS|D) = P(POS)? P(POS|D) =P(POS and D)/P(D) = 0.009/0.010 = 0.90 P(POS) = 0.256 The events POS and D are dependent Agresti/Franklin Statistics, 31 of 87 Section 5.4 Applying the Probability Rules Agresti/Franklin Statistics, 32 of 87 Is a “Coincidence” Truly an Unusual Event? The law of very large numbers states that if something has a very large number of opportunities to happen, occasionally it will happen, even if it seems highly unusual Agresti/Franklin Statistics, 33 of 87 Example: Is a Matching Birthday Surprising? What is the probability that at least two students in a group of 25 students have the same birthday? Agresti/Franklin Statistics, 34 of 87 Example: Is a Matching Birthday Surprising? P(at least one match) = 1 – P(no matches) Agresti/Franklin Statistics, 35 of 87 Example: Is a Matching Birthday Surprising? P(no matches) = P(students 1 and 2 and 3 …and 25 have different birthdays) Agresti/Franklin Statistics, 36 of 87 Example: Is a Matching Birthday Surprising? P(no matches) = (365/365) x (364/365) x (363/365) x … x (341/365) P(no matches) = 0.43 Agresti/Franklin Statistics, 37 of 87 Example: Is a Matching Birthday Surprising? P(at least one match) = 1 – P(no matches) = 1 – 0.43 = 0.57 Agresti/Franklin Statistics, 38 of 87