条件概率小题目

Question：
Suppose a family has children, one of which is a boy. What is the probability that both children are boys?

A)1 / 3
B)1 / 2
C)2 / 3
D)1 / 9

这种题目真的会把人绕晕，解答：
1/2：理解为第一个是男孩,第二个也是男孩的概率,两个概率相互独立,所以为1/2
1/3：理解为已知两人中有一人是男孩,但不知道是第一个还是第二个,即两人中至少有一名男孩（概率为3/4）,利用条件概率公式,求得另一人为男孩的概率是P(A|B)=P(AB)/P(B)=(1/4) / (3/4 ) = 1/3
因为题目上是某一个,所以应该是1/3，选择A

详解：设A事件为“另一个孩子也是男孩”，P(AB）=1/4
设B事件为“两个孩子中有一个男孩”，P(B)=1-1/4=3/4

Conditional Probability
This is defined as the probability of an event occurring, assuming that one or more other events have already occurred. Two events, A and B are considered to be independent if A event has no effect on the probability of event B (i.e. P(B|A)=P(B)). If events A and B are not independent, then we must consider the probability that both events occur. This can be referred to as the intersection of events A and B , defined as P(AB)=P(B|A)P(A). We can then use this definition to find the conditional probability by dividing the probability of the intersection of the two events (AB) by the probability of the event that is assumed to have already occurred (event A):

Bayes’ Theorem
Let A and B be two events such that P(A|B) denotes the probability of the occurrence of A given that B has occurred and P(B|A) denotes the probability of the occurrence of B given that A has occurred, then: