Probability of Even vs. Odd Sums

Date: 10/11/2000 at 17:32:15
From: Anonymous
Subject: Probability of even vs. odd sums 

Hi Dr. Math,

My second-grade son asked a question of his math teacher and was not 
satisfied with her answer. She taught the class that the sum of two 
even numbers will be even, the sum of two odd numbers will be even, 
and the sum of one odd and one even number will be odd.

He asked her, since there are more ways to achieve an even sum, is it 
more likely that an addition problem will have an even answer? She 
said, "No, it depends on the addends." He understands that it depends 
on the specific problems he is given, but still feels that overall, 
addition problems are more likely to have even answers.

Can you provide a reasonable explanation for him as to why this is 
true or not true?

Thank you in advance

Date: 10/12/2000 at 12:15:44
From: Doctor Anthony
Subject: Re: Probability of even vs. odd sums 

You have to be a little careful here because you are dealing with 
infinite sets. However, if we are limited to the set of numbers 1 to 
100, there are 50 even numbers and 50 odd numbers.

   We can choose 2 even numbers in 50 x 50   = 2500 ways
   We can choose 2 odd numbers in 50 x 50    = 2500 ways
   We can choose 1 odd and 1 even in 50 x 50 = 2500 ways
   We can choose 1 even and 1 odd in 50 x 50 = 2500 ways

The sum of the two numbers is even on 2 x 2500 = 5000 occasions. The 
sum of the two numbers is odd on 2 x 2500 = 5000 occasions. So there 
is no bias in favor of an even sum.

- Doctor Anthony, The Math Forum
  http://mathforum.org/dr.math/