The ELISA test was an early test used to screen blood...

The ELISA test was an early test used to screen blood donations for antibodies to HIV. A study (Weiss et. al. 1985) found that the conditional probability that a person would test positive given they have HIV was 0.97 and the conditional probability that a person would test negative given they did NOT have HIV was 0.926. The World Almanac gives an estimate of the probability of a person in the USA of having HIV to be 0.0026. Question #16 Suppose a random person is tested and they test positive. What is the conditional probability that this person has HIV given that they test positive? Question #17 Are you surprised by your answer? What implications does this have for policies of mandatory testing? Note that this phenomenon of large and unexpected changes in conditional probabilities is not unusual, particularly when dealing with rare events. What is happening is that the number of false positives is much larger than the number of true positives. Question #18 Suppose 10000 random people are tested. How many of them do you expect to actually have HIV? Question #19 Of those with HIV, how many do you expect to test positive? Question #20 Of those without HIV, how many do you expect to test positive? Question #21 Do your answers to Question #19 and #20 make sense in light of your answer to #16?

Transcript

00:01 Hi, i'm david and i'm here you have you answered your question.

00:03 In the question here let me summarize the question by the notation.

00:07 Here i will denote the test constituted by the v, stand for the positive test and then i divide the arch stand for the person who will have the hv.

00:28 And then in the question here we are given the description that the probability that that a person would test positive given that they have an hiv equal to 097 and then the next one they told us that the conditional probability that the person would test negative i will call this one and stand for the negative test and then we will have probability of the negative test given that the person did not have the hiv would be the archgram here we stand for not having the hiv equal to 0 .9 to 6 and we also told that the probability and the person have the hiv equal to 0 .026 and now let me draw the vein the three diagram to illustrate this is and be easier so this will be the hiv we will be the hiv we will be not having the hiv, the chance to have the hiv equal to 0 .006.

01:50 And not having the hiv will be 1 minus 0 .0 .0 .26 equal to the 0 .9 .74.

02:01 And then we'll turn the test and then the result will be the positive or the negative.

02:05 Positive or the negative.

02:09 And however achieving the test positive will be 097.

02:15 The common 1 minus 097 equals to the zabon 0 3.

02:19 And this one will be and the test is negative here will be 0 .926 and 1 minus 0 .9 26 equal to 0 .04 and that will be the summary of the question.

02:32 Now we are ready to answer the question 16 now.

02:35 Now, if the random person is tested and they test positive, so probability given that the test is constitute.

02:46 What is the probability that this person has the hiv, so hiv given the test is positive? so means that this one will be by the conditional probability equal to probability in the hiv and the positive test development of proper pretty than the positive test and we will find the denominator by going to the hiv then go to the positive although children not have an hiv then go to the positive then we will just multiply them like a pair then we will have on the bottom will mean the 0 .026 then with the 0 .97 plus 0 .994 10 when the 0 .077 on the top, we want to have the arch .a .v and the positive will be this frame here.

03:41 Then we will have on the top will be the 0 .0026 times under 0 .97.

03:49 And if we compute it, we have 0 .0026 times 0 .97.

03:55 Defined by the answer first, 099 -74 times 0 .0 .74 equal to the 0 .0.

04:04 One do three three and then the next one we will have the question 17 are you surprised by your answer what indication does this here for the policy of mandatory testing so yes we are yes we are surprised yes we are surprised and because this probability here very small because this probability of the heavy the negative and having the hiv when we got the the ponstil test is only equal to the 3 .3 percent so it means that even when we get the font still result it doesn't mean that we will get the hiv and the chance to get the archiv only 3 .3 percent which is very small and then we have the next question 18 suppose then the 1 10 ,000 random people outlasted so here in the question 18 we have the n equal to the 10 ,000 people now how many of them do you expect to actually have the hiv so having the hiv so we have the probability of the hiv equal to the 0 -026 so the expected number having hiv equal to the 0 .026 time window 10 ,000.

06:10 Then we have your each other equal to the 26 people.

06:16 And then we have the question 19...

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