AMC10 2012 B

AMC10 2012 B · Q18

AMC10 2012 B · Q18. It mainly tests Probability (basic), Conditional probability (basic).

Suppose that one of every 500 people in a certain population has a particular disease, which displays no symptoms. A blood test is available for screening for this disease. For a person who has this disease, the test always turns out positive. For a person who does not have the disease, however, there is a 2% false positive rate—in other words, for such people, 98% of the time the test will turn out negative, but 2% of the time the test will turn out positive and will incorrectly indicate that the person has the disease. Let p be the probability that a person who is chosen at random from this population and gets a positive test result actually has the disease. Which of the following is closest to p ?

假设某人群中每500人中有一人患有某种无症状疾病。有一种血液检测可用于筛查该疾病。对于患病者，检测总是阳性。对于未患病者，假阳性率为2%——即98%情况下检测阴性，2%情况下检测阳性，错误指示患病。设p为从该人群随机选中一人并检测阳性后实际患病的概率。下列哪项最接近p？

(A) 1/98 1/98

(B) 1/9 1/9

(D) 49/99 49/99

(E) 98/99 98/99

Answer

Correct choice: (C)

正确答案：（C）

Solution

On average for every 500 people tested, 1 will test positive because he or she has the disease, while 2% · 499 ≈ 10 will test positive even though they do not have the disease. In other words, of approximately 11 people who test positive, only 1 has the disease, so the probability is approximately$\frac{1}{11}$.

平均每500人检测，1人因患病呈阳性，而2% · 499 ≈ 10人虽未患病但呈阳性。也就是说，大约11人呈阳性中，只有1人患病，故概率约为$\frac{1}{11}$。

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