3.- Two percent of the USA population suffers from the Donald Duck's disease. The Center for Disease Control has a test to diagnose for this disease. The test is 99% effective if the person has the disease and 98% effective if the person does not have the disease. What is the probability that a randomly selected person taking the test: (a) will have a "Positive" result? (b) will have a "Negative" result? (c) Given that the person tested positive, what is the probability that he/she actually is infected with this disease?
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Let P be the event that the test result is "Positive", and N be the event that the test result is "Negative". From the problem statement, we are given the following probabilities: P(D) = 0.02 (Two percent of the USA population suffers from the disease) P(D') = 1 Show more…
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a) A laboratory blood test is 95% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a "false positive" result for 1% of the healthy persons tested. (That is, if a healthy person is tested, then, with a probability of 0.01, the test result will imply he or she has the disease.) If 0.5% of the population actually has the disease, what is the probability a person has the disease given that the test result is positive? b) What is the probability that a randomly selected person, who has the disease, gets a positive result from the blood test? Explain the difference between this probability and the probability you calculated in a). c) Are the two events that a person has the disease and that the blood test is positive dependent? Does the fact that a test result was positive increase the risk of having the disease, compared to the probability that a random individual from the population has the disease?
Adi S.
A diagnostic test for a disease returns one of two answers: Positive (has the disease), negative (does not have the disease). Here are the probabilities: • For a patient with the disease, 95% positive / 5% negative • For a patient that does not have the disease, 20% positive / 80% negative (a) Suppose that 5% of the population has the disease. If everyone is tested, what proportion of the test results will be positive? (b) Suppose again that 5% of the population has the disease. For a patient who gets a positive result, what is the probability of having the disease? (c) Now, let d be the prevalence of the disease in the population (for example, if 5% of the population has the disease as above, then d = 0.05). First, everyone in the population is screened for the disease. For each person that got a positive result, they are brought back for a second screening. It turns out that 25% of the return patients test positive the second time. (You can assume that, for any single patient, their first test result and their second test result are independent—for example, if I have the disease and get a positive on the first test, the second time I take the test it’s a fresh "roll of the dice" with the same chances of getting any of the two possible results.) What is d? You can assume that it’s a large population so all the probabilities work out exactly as expected (e.g. if you flip n fair coins then assume exactly 0.5 * n of them are Heads, etc).
Twee M.
Assume that 0.4% of the population has a condition that is not detectible by simple external observation. A diagnostic test is available for this condition, but, like most tests, it is not perfect. The test correctly diagnoses, with a positive result, those with the condition 99.7% of the time. The test correctly identifies, with a negative result, those without the condition 98.5% of the time. Let the event C1 represent the presence of the condition and C2 represent the absence of the condition, and let event T represent a positive test result (meaning the test indicated, either correctly or incorrectly, that a person has the condition) and TC represent a negative test result (meaning the test indicated, either correctly or incorrectly, that a person does not have the condition). List symbolically the information given in the introductory statement at the top of the page. If a randomly selected member of the population tests positive for the condition, what is the probability that the person has the condition? (Report or round your answer to 3 decimal places.) If a randomly selected member of the population tests positive for the condition, what is the probability that the person does not have the condition? If a randomly selected member of the population tests negative for the condition, what is the probability that the person has the condition? (Report or round your answer to 6 decimal places.)
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