A test for a certain disease is found to be 95% accurate, meaning that it will c

Question

A test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out of 100 people who have the ailment. The test is also 95% accurate for a negative result, meaning that it will correctly exclude the disease in 95 out of 100 people who do not have the ailment. For a certain segment of the population, the incidence of the disease is 4%. (1) If a person tests positive, find the probability that the person actually has the disease (Hint: define appropriate events and use Bayes Theorem); (2) Now suppose the incidence of the disease is 49%. Compute the probability that the person actually has the disease, given that the test is positive; (3) The probability you obtained in (1) is much smaller than 0.95, if your computation is correct. Hence, You can conclude that there is only a much smaller probability to claim “the person really has the disease” after knowing that “the test is positive”, though the test has 95% “correctness”. Explain this difference. A test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out of 100 people who have the ailment. The test is also 95% accurate for a negative result, meaning that it will correctly exclude the disease in 95 out of 100 people who do not have the ailment. For a certain segment of the population, the incidence of the disease is 4%. (1) If a person tests positive, find the probability that the person actually has the disease (Hint: define appropriate events and use Bayes Theorem); (2) Now suppose the incidence of the disease is 49%. Compute the probability that the person actually has the disease, given that the test is positive; (3) The probability you obtained in (1) is much smaller than 0.95, if your computation is correct. Hence, You can conclude that there is only a much smaller probability to claim “the person really has the disease” after knowing that “the test is positive”, though the test has 95% “correctness”. Explain this difference. A test for a certain disease is found to be 95% accurate, meaning that it will correctly diagnose the disease in 95 out of 100 people who have the ailment. The test is also 95% accurate for a negative result, meaning that it will correctly exclude the disease in 95 out of 100 people who do not have the ailment. For a certain segment of the population, the incidence of the disease is 4%. (1) If a person tests positive, find the probability that the person actually has the disease (Hint: define appropriate events and use Bayes Theorem); (2) Now suppose the incidence of the disease is 49%. Compute the probability that the person actually has the disease, given that the test is positive; (3) The probability you obtained in (1) is much smaller than 0.95, if your computation is correct. Hence, You can conclude that there is only a much smaller probability to claim “the person really has the disease” after knowing that “the test is positive”, though the test has 95% “correctness”. Explain this difference.

Explanation / Answer

dear student please post the question one at a time

1)The following means the probability of getting a positive result, given that you are positive:
p(+ve result | are +ve ) = 0.95 = 95%. we are given this
p(-ve result | are +ve ) = 0.05 = 5%. we are given this
p(-ve result | are -ve ) = 0.95 = 95%. we are given this
p(+ve result | are -ve ) = 0.05 = 5%. we are given this

p(+ve result) = p(+ve)*p(+ve result | are +ve) + p(-ve)*p(+ve result | are -ve ) = 0.04 * 0.95 + 0.96 * 0.05 = 0.086 = 8.6%

we want the prob. that you are positive given you tested positive.
p(are +ve | +ve result) = p(+ve result | are +ve ) * p(are +ve) / p(+ve result)
= 0.95 * 0.04 / 0.086
= 0.44 = 44%

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