Statıstıcs I Deneme Sınavı Sorusu #836463
Consider an ultrasound software for brain tumor diagnosis : a) the overall rate of the disease in the population being screened is 1 % ; b) the probability that a healthy person wrongly gets a positive result (false positive) is 0.05 ; c) the probability that an ill wrongly gets a negative result (false negative) is 0.002 ; d) other 2 situations are correctly diagnosing healthy persons (true negative) and correctly diagnosing ills (true positive). If test of a person A gives a positive result, what is the probability that person A actually have the disease?
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0.0495 |
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0.998 * 0.01 / 0.05948 |
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0.06 |
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0.099 * 0.02 / 0.064 |
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0.095 |
S = {people in the population being screened} , D = {have disease} , not-D = {do not have diesase} , pos = {positive result} , neg = {negative result} ; P (pos | not-D) = 0.05 , P (neg | D) = 0.002 , P (D) = 0.01 ; P (D | pos) = ? , P (D | pos) = P (pos | D) * P(D) / P(pos) (Bayes theorem) ; P (pos | D) = 1 - P ( neg | D) = 1 - 0.002 = 0.998 ; P (pos) = ? , P (pos) = P (pos | D) * P(D) + P (pos | not-D) * P(not-D) ; P (pos) = 0.998 * 0.01 + 0.05 * (1 - 0.01) = 0.00998 + 0.05 x 0.99 = 0.00998 + 0.0495 = 0.05948 ; P (D | pos) = 0.998 * 0.01 / 0.05948 ( = 0.168 = 16.8 % ) . pg. 141. Correct answer is B.
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