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Section 5.3

Section 5.3. Conditional Probability: What’s the Probability of A, Given B?. Conditional Probability. For events A and B, the conditional probability of event A, given that event B has occurred is:. Conditional Probability. Example: What are the Chances of a Taxpayer being Audited?.

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Section 5.3

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  1. Section 5.3 Conditional Probability: What’s the Probability of A, Given B?

  2. Conditional Probability • For events A and B, the conditional probability of event A, given that event B has occurred is:

  3. Conditional Probability

  4. Example: What are the Chances of a Taxpayer being Audited?

  5. Example: Probabilities of a Taxpayer Being Audited

  6. Example: Probabilities of a Taxpayer Being Audited • What was the probability of being audited, given that the income was ≥ $100,000? • Event A: Taxpayer is audited • Event B: Taxpayer’s income ≥ $100,000

  7. Example: Probabilities of a Taxpayer Being Audited

  8. 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

  9. 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

  10. 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

  11. Example: The Triple Blood Test for Down Syndrome

  12. 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

  13. Example: The Triple Blood Test for Down Syndrome • P(POS) = 1355/5282 = 0.257

  14. 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

  15. Example: The Triple Blood Test for Down Syndrome

  16. Example: The Triple Blood Test for Down Syndrome • Summary: Of the women who tested positive, fewer than 4% actually had fetuses with Down syndrome

  17. 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)

  18. 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

  19. 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?

  20. 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

  21. Sampling Without Replacement • Once subjects are selected from a population, they are not eligible to be selected again

  22. 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?

  23. 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)

  24. 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

  25. 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

  26. 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)

  27. 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

  28. 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

  29. Example: How to Check Whether Two Events are Independent Blood Test:

  30. Example: How to Check Whether Two Events are Independent • Are the events POS and D independent or dependent? • Is P(POS|D) = P(POS)?

  31. 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

  32. Section 5.4 Applying the Probability Rules

  33. 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

  34. 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?

  35. Example: Is a Matching Birthday Surprising? • P(at least one match) = 1 – P(no matches)

  36. Example: Is a Matching Birthday Surprising? • P(no matches) = P(students 1 and 2 and 3 …and 25 have different birthdays)

  37. 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

  38. Example: Is a Matching Birthday Surprising? • P(at least one match) = 1 – P(no matches) = 1 – 0.43 = 0.57

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