A test for Covid-19 gives either positive or negative...

A test for Covid-19 gives either positive or negative results, but it is not 100% reliable. In other words, there may be false positives and false negatives. A probability tree diagram for testing a person, some events and probabilities are given below. Give the probabilities for each branch, paths (indicated by small letters a through j both on the diagram and also in the question parts below) and additional parts k and l). Remark: Round the probabilities to two digits after the decimal. Stage1: Condition Stage 2: a b c d e f g Has virus and tests negative h i Does not have virus and tests negative j V: Person has the Covid-19 virus. N: Test results negative. P: Test results positive. p(V) = 0.31 p(N | V) = 0.14 p(N | V') = 0.92 a. c. e. g. i. k. P(V | N) = b. d. f. h. j. l. P(N) = Note: a, b, c, e, l are 1 point each, k is 3 points, and the rest are 2 points each.

Transcript

00:01 All right.

00:02 So we're being asked to fill out a probability tree for a covid test where the first outcome is that the person has a virus with test negative, second that the virus has a virus and test positive, and so on, as is written here.

00:18 And we're given that the probability that a person has a virus is 0 .31, and the probability that given a person has a virus, they test negative is 0 .14.

00:28 And the probability that given a person doesn't have the virus and they test negative is 0 .92.

00:34 So first, we need to figure out what the first branch and the second branch are each deciding.

00:40 And since both of the top outcomes have a virus and both of the top bottom outcomes have no virus, we know that this means that the person has a virus, and this branch means a person doesn't have the virus.

00:56 And now because the second branch, always just negative at the top, this means they test negative, and this means that they test positive.

01:12 So the first branch is basically saying, what is the probability that a person has a virus? we're given that this is equal to 0 .31, and so you can either have the virus or not have the virus, so the probability that you don't have the virus is equal to 1, minus 0 .31, which is equal to 0 .69.

01:31 Now, the second branches all represent the conditional probabilities, because we're already assuming the case here, that the person has the virus.

01:40 And so this probability is equal to the conditional probability that given the person has the virus, they test negative.

01:46 This is also given to us at 0 .14.

01:51 Similarly, on the condition that i'm assuming the person has already shown to not have the virus, the probability that they test negative is 0 .92.

02:08 Now, because these are already assuming that the person either has a virus or not, the conditional of probabilities also have to sum up to 1.

02:16 So in this case, 1 minus 0 .14 gives us a conditional probability that you test positive if you have the virus of 0 .86.

02:30 And 1 minus 0 .92 gives us a conditional probability that you test positive if you don't have a virus of 0 .08.

02:40 So finally, to get our n probabilities, we multiply and we get 0 .314 times 0 .14, means the probability that a person both has a virus and test negative is equal to 0 .04 to 2 decimal points...

Question

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