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Section 7.6 Bayes’ Theorem and Applications We can also relate tree diagrams to having a disease and how a person/item tested. If they have a disease, does the test also support that result? New terms: Testing Positive: means that the test thinks that the person/item has the disease. Testing Negative: means that the test thinks that the person/item does not have the disease. Test Condition Has the Disease Doesn’t have the Disease Positive Negative Event True Positive False Negative Positive False Positive Negative True Negative Four Disease Conditions True Positive: The test states that the person/item has the disease when they do really have the disease False Negative: The test states that the person/item does not have the disease when they do have the disease False Positive: The test states that the person/item has the disease when they really do not have the disease True Negative: The test states that the person/item does not have the disease when they do not have the disease Example 1: Problem 3.2.7 from Statistics for Life Sciences Suppose that a medical test has a 92% chance of detecting a disease if the person has the disease and a 94% chance of correctly indicating that the disease is absent if the person really does not have the disease. Suppose that 10% of the population has the disease. 1. What is the probability that a randomly chosen person will test positive? 2. Suppose that a randomly chosen person does test positive. What is the probability that this person really has the disease? Let D be the event that the person has the disease. Let P be the event that the person tests positive. We will need to use some equations that we have learned about this previously P( A B) P( A) P ( B A) or P( A B) P( B) P( A B) Pr A B Pr(B) Pr( A and B) and PrB A Pr( A) Pr( A and B) P( B) P( B A) P( A) Pr(B A ) Pr( A ) This problem illustrates the use of Bayes’ Theorem and Applications: Bayes’ Theorem (Short Form) If A and T are events, then Bayes Formula P(T A) P( A) P( A T ) P(T A) P( A) P(T A) P( A) P( A T ) P( A T ) P( A T ) P( A T ) Using a Tree P( A T ) P( Using A and T branches) Sum of P( Using branches ending in T ) Example 2 Use Bayes’ theorem or a tree diagram to calculate the indicated probability. 3. P( X Y ) 0.8 , P (Y ) 0.3, P( X Y ) 0.5 , Find P (Y X ) Bayes’ Theorem (Expanded Form) If the events A1, A2 , and A3 form a partition of the sample space S , then P(T A1 ) P( A1 ) P( A1 T ) or P(T A1 ) P( A1 ) P(T A2 ) P( A2 ) P(T A3 ) P( A3 ) P( A1 T ) P( A1 T ) P( A1 T ) P( A2 T ) P( A3 T ) Example 3 Use Bayes’ theorem or a tree diagram to calculate the indicated probability. 3. Y1, Y2 , Y3 form a partition of S . P( X Y1 ) 0.2 , P( X Y2 ) 0.3 , P( X Y3 ) 0.6 , P(Y1 ) 0.3 , P(Y2 ) 0.4 , Find P (Y1 X ) 18. Professor Frank Nabarro insists that all senior physics majors take his notorious physics aptitude test. The test is so tough that anyone not going on to a career in physics has no hope of passing, whereas 60% of the seniors who do go on to a career in physics still fail the test. Further, 75% of all senior physics majors in fact go on to a career in physics. Assuming that you fail the test, what is the probability that you will not go on to a career in physics? 24. In 2000, 59% of all Caucasians in the U.S., 57% of all AfricanAmericans, 58% of all Hispanics, and 54% of residents not classified into one of these groups used the Internet to search for information. At that time, the U.S. population was 69% Caucasian, 12% African-American, and 13% Hispanic. What percentage of U.S. residents who used the Internet for information search were African-American? 26. A New York Times survey showed that 51% of those who had no preschool education were arrested or charged with a crime by the time they were 19, whereas only 31% who had preschool education wound up in this category. The survey did not specify what percentage of the youths in the survey had preschool education, so let us take a guess at that and estimate that 20% of them had attended preschool. What percentage of the youths arrested or charged with a crime had no preschool education?