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Calculating Conditional and Unconditional Probabilities

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The probability of having disease X in the general population is only .05. The Sagman Test is a newly discovered method for early detection. Of those who have disease X, the test indicates the disease for 90% of them. Of those who do not have the disease, the test indicates no disease for 90% of them. Is the test a good predictor of whether you actually have the disease or not? Should its use be widespread?

a. Which of the following conditional probabilities is most appropriate to determine the test's effectiveness? Explain.

P(have disease X / test says you have the disease)
P(test says you have the disease / have the disease)
P(test says you have the disease)

b. Compare the unconditional probability of having the disease to the conditional probability of having the disease given the test says you have the disease.

c. Compare the unconditional probability of not having the disease to the conditional probability not having the disease given the test says you do not have the disease.

d. Based on parts b and c, is the test a good predictor? Explain.

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The solution helps with questions that involve conditional and unconditional probability calculations.

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The probability of having disease X in the general population is only .05. The Sagman Test is a newly discovered method for early detection. Of those who have disease X, the test indicates the disease for 90% of them. Of those who do not have the disease, the test indicates no disease for 90% of them. Is the test a good predictor of whether you actually have the disease or not? Should its use be widespread?

Conditional probability is the probability of some event A, given the occurrence of some other event B. Conditional probability is written P(A|B), and is read "the (conditional) probability of A, given B" or "the probability of A under the condition B". When in a random experiment the event B is known to have occurred, the possible outcomes of the experiment are reduced to B, and hence the probability of the occurrence of A is changed from the unconditional probability into the conditional probability given B.

The conditional probability can be calculated as P(A | B) = P(A and B)/ P(B)

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a. Which of the following conditional probabilities is most appropriate to determine the test's effectiveness? Explain.
P(have disease X / test says you have the disease) ...

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