The Solution of a Mathematical Probability Problem

Let Ai= "The I-th lid just covers the I-th cup", i= 1, 2, ..., 5.

Let B= "At least one lid is matched with the cup"

Then B=A 1+A2+...+A5.

P(A 1+A2+...+A5)

=P(A 1)+P(A2)+...+P(A5)

-P(A 1A2)-P(A 1A3)-...-P(A4A5)

+P(a 1a2 a3)-P(a 1a2 a4)+...+P(A3A4A5)-

...

+P(A 1A2A3A4A5)

Because there are five ways to match the lid! A little, so

P(a 1)= P(A2)= 1...=P(A5)=4！ /5!

P(a 1 a2)= P(a 1 a3)= P...=P(A4A5)=3！ /5!

...

P(A 1A2A3A4A5)= 1/5！

therefore

P(B)

=P(A 1+A2+...+A5)

=(4! /5! )*5-(3! /5! )*C(5，2)+(2！ /5! )*C(5，3)-...+ 1/5!

P (all errors) = 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!