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Returning Homework and Probability

Date: 12/11/98 at 15:13:40
From: Adam Bergman
Subject: Probability

I'm passing back 25 papers to my 4th grade math challenge class. It's 
the homework that they just brought me, and I decided that we would 
grade each other's papers. One of the students raised her hand and 
said, "What if we get our own paper back?" I offer extra credit to 
anyone who does, since I am passing them out completely randomly.  
Another child says "Hey, that's not fair. The kids at the end have less 
chance of getting their own papers back." Another shouts "No, it's 
the other way around. The people at the beginning have less chance of 
winning!" I say "No, everyone has a 1/25 chance of getting his or her 
own paper back." 

The only thing is that I cannot explain why. The first one is easy. One 
winner out of 25 possibilities. But, how about the others?

Date: 12/11/98 at 18:06:58
From: Doctor Anthony
Subject: Re: Probability

The chance that the first child will get the correct paper is as you 
say: 1/25. For the second child there are 24 papers, so the chance of 
getting the correct one is:

   first child correct, second child correct + first child incorrect   
   (but not second child's paper), second child correct.

So the probability is:

   Prob =  1/25 x 1/24 + 23/25 x 1/24
        = (1/25)[1/24 + 23/24]
        = (1/25)[1]
        = 1/25

and so on.

An easier explanation is as follows. There is no change to the 
probability situation if all the papers are placed round a table face 
down. All the children take up positions round the table at random.  
There is no reason for any child to be favoured in any way in this 
scenario. Finally, all together they turn over the paper in front of 
them. They all have the same chance that they have their own paper.

- Doctor Anthony, The Math Forum   
Associated Topics:
High School Probability

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