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Chapters 14, 15 (part 2) Probability Trees, Odds i)Probability Trees: A Graphical Method for Complicated Probability Problems. ii)Odds and Probabilities

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Probability Tree Example: probability of playing professional baseball 6.1% of high school baseball players play college baseball. Of these, 9.4% will play professionally. Unlike football and basketball, high school players can also go directly to professional baseball without playing in college… studies have shown that given that a high school player does not compete in college, the probability he plays professionally is.002. Question 1: What is the probability that a high school baseball player ultimately plays professional baseball? Question 2: Given that a high school baseball player played professionally, what is the probability he played in college?

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Question 1: What is the probability that a high school baseball player ultimately plays professional baseball P(hs bb player plays professionally) =.061*.094 +.939*.002 =.005734 +.001878 =.007612.061*.094=.005734.939*.002=.001878

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Question 2: Given that a high school baseball player played professionally, what is the probability he played in college?.061*.094=.005734.939*.002=.001878 P(hs bb player plays professionally) =.005734 +.001878 =.007612.061*.094=.005734

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Example: AIDS Testing zV={person has HIV}; CDC: Pr(V)=.006 zP : test outcome is positive (test indicates HIV present) zN : test outcome is negative zclinical reliabilities for a new HIV test: 1.If a person has the virus, the test result will be positive with probability.999 2.If a person does not have the virus, the test result will be negative with probability.990

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Question 1 zWhat is the probability that a randomly selected person will test positive?

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Probability Tree Approach zA probability tree is a useful way to visualize this problem and to find the desired probability.

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Probability Tree Multiply branch probs clinical reliability

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Question 1: What is the probability that a randomly selected person will test positive?

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Question 2 zIf your test comes back positive, what is the probability that you have HIV? (Remember: we know that if a person has the virus, the test result will be positive with probability.999; if a person does not have the virus, the test result will be negative with probability.990). zLooks very reliable

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Question 2: If your test comes back positive, what is the probability that you have HIV?

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Summary zQuestion 1: zPr(P ) =.00599 +.00994 =.01593 zQuestion 2: two sequences of branches lead to positive test; only 1 sequence represented people who have HIV. Pr(person has HIV given that test is positive) =.00599/(.00599+.00994) =.376

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Recap zWe have a test with very high clinical reliabilities: 1.If a person has the virus, the test result will be positive with probability.999 2.If a person does not have the virus, the test result will be negative with probability.990 zBut we have extremely poor performance when the test is positive: Pr(person has HIV given that test is positive) =.376 zIn other words, 62.4% of the positives are false positives! Why? zWhen the characteristic the test is looking for is rare, most positives will be false.

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ODDS AND PROBABILITIES zWorld Series OddsWorld Series Odds zFrom probability to odds zFrom odds to probability

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From Probability to Odds zIf event A has probability P(A), then the odds in favor of A are P(A) to 1-P(A). It follows that the odds against A are 1-P(A) to P(A) z If the probability the Boston Red Sox win the World Series is.20, then the odds in favor of Boston winning the World Series are.20 to.80 or 1 to 4. The odds against Boston winning are.80 to.20 or 4 to 1

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From Odds to Probability zIf the odds in favor of an event E are a to b, then P(E)=a/(a+b) zIf the odds against an event E are c to d, then P(E’)=c/(c+d) (E’ denotes the complement of E) Team Odds against winning P(E’)=Prob of not winning RED SOX4/14/5=.80 DODGERS5/15/6=.833 TIGERS5/15/6=.833 CARDINALS11/211/13=.846 BRAVES7/17/8=.875 A’s15/215/17=.882 TB RAYS14/114/15=.933 INDIANS14/114/15=.933 REDS16/116/17=.941 PIRATES16/116/17=.941 E = win World Series

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4-4 Multiplication Rule The basic multiplication rule is used for finding P(A and B), the probability that event A occurs in a first trial and event B.

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