math - 排列与组合面试

Question

这是一个很好的选择，因为它与直觉相反:

Imagine an urn filled with balls, two-thirds of which are of one color and one-third of which are of another. One individual has drawn 5 balls from the urn and found that 4 are red and 1 is white. Another individual has drawn 20 balls and found that 12 are red and 8 are white. Which of the two individuals should feel more confident that the urn contains two-thirds red balls and one-third white balls, rather than vice-versa? What odds should each individual give?

我知道正确的答案，但也许我不太了解赔率计算。谁能解释？

Answer 1

令 A 是2/3的球是红色的事件，然后¬ A 是2/3的球是白色的事件。假设 B 是第一个观察者看到5个红球的事件，而 C 是第二个观察者看到20个红球的事件。

应用一些简单的组合函数，我们得到

P( B | A )=(5选择4)(2/3)4(1/3)1 = 80/243

P( B |¬ A )=(5选择4)(1/3)4(2/3)1 = 10/243

因此，根据贝叶斯定律，观察者1的置信度为80 /(80 + 10)= 8/9， 为真。

对于第二个观察者:

P( C | A )=(20选择12)(2/3)12(1/3)8 = 125970 * 212/320

P( C |¬ A )=(20选择12)(1/3)12(2/3)8 = 125970 * 28/320

因此，再次根据贝叶斯定律，观察者2的置信度为212 /(212 + 28)= 16/17， 是真实的。

因此， 观察者两个具有较高的置信度，即2/3的球是红色的。关键是要了解贝叶斯定律的工作原理。实际上，重要的是观察到的红色和白色球的数量不同。其他所有内容(特别是抽出的球总数)在方程式中抵消。