2012 AMC 10B 第 18 题

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

小提示：

每

500

人中，大约有

1

个真阳性和

10

个假阳性。

Out of

500

people, about

1

true positive and

10

false positives occur

大提示：

在检测结果为阳性的条件下计算概率。

Condition on the test being positive

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

小提示：

大提示：

解答：

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