Question

假设某人群中每 $500$ 人有一人患有某种无症状疾病。有一种血液检测可用于筛查这种疾病。患病者检测结果一定为阳性。

对于没有患病的人，检测有 $2\%$ 的假阳性率。也就是说，这些人中 $98\%$ 的检测结果为阴性，但 $2\%$ 的检测结果会误为阳性。

令 $p$ 为从该人群中随机选一人，且检测结果为阳性时，此人实际患病的概率。下列哪一个最接近 $p$ ？

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 $p?$

$\dfrac{1}{98}$

$\dfrac{1}{9}$

$\dfrac{1}{11}$

$\dfrac{49}{99}$

$\dfrac{98}{99}$

答案：C

知识点：条件概率贝叶斯定理

难度评级：1730

解答：

在 $500$ 人中，大约 $1$ 人患病并检测为阳性。其余 $499$ 人中约有 $2\%$ 假阳性，也就是约 $10$ 人。

因此约 $11$ 个阳性结果中，只有约 $1$ 个是真阳性，概率最接近 $\dfrac1{11}$ 。

所以正确答案是 C。

Among $500$ people, about $1$ person has the disease and tests positive. Of the remaining $499$ people, about $2\%$ test falsely positive, which is about $10$ people.

So among about $11$ positive tests, only about $1$ is a true positive. The probability is closest to $\dfrac1{11}$ .

Thus, C is the correct answer.

2012 AMC 10B 第 18 题

答案：C

难度评级：1730

解答：

其他年份的第 18 题