Question

一个数学概率问题求解

五个茶杯相对应五个盖子，随意将它们搭配在一起，求一次全部错误的概率（要过程！）

Answer 1

设Ai=“第i个盖子正好盖在第i个杯子上”，i=1,2，...，5
设B=“至少有一个盖子和杯子配对”
则B=A1+A2+...+A5
P(A1+A2+...+A5)
=P(A1)+P(A2)+...+P(A5)
-P(A1A2)-P(A1A3)-...-P(A4A5)
+P(A1A2A3)-P(A1A2A4)+...+P(A3A4A5)-
...
+P(A1A2A3A4A5)
由于杯子跟盖子配对的方式有5！种，所以
P(A1)=P(A2)=...=P(A5）=4！/5！
P(A1A2)=P(A1A3)=...=P(A4A5)=3!/5!
...
P(A1A2A3A4A5)=1/5!
于是
P(B)
=P(A1+A2+...+A5)
=(4!/5!)*5-(3!/5!)*C(5,2)+(2!/5!)*C(5,3)-...+1/5!
P(全部错误)=1-PB)
=1-(4!/5!)*5+(3!/5!)*C(5,2)-(2!/5!)*C(5,3)+...-1/5!
=(3!/5!)*C(5,2)-(2!/5!)*C(5,3)+...-1/5!

Answer 2

得（4/5）*(3/4)*(2/3)*(1/2)=1/5

Answer 3

4*4*3*2*1/120=4/5