SOLUTION: A box contains ten balls numbered from 1 to 10. If you pick two balls at random, what is the probability that the sum of the numbers on the two balls is even?

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Question 323342: A box contains ten balls numbered from 1 to 10. If you pick two balls at random, what is the probability that the sum of the numbers on the two balls is even?

Found 2 solutions by galactus, Alan3354:
Answer by galactus(183) (Show Source):
You can put this solution on YOUR website!
There are 2 balls chosen. Thus, there are 10^2 = 100 possible outcomes.
Half of them sum to odds and half to evens.
Perhaps you have not dealt with generating functions, but we can illustrate it with them.

If we expand this out, we get

The coefficients represent the number of ways that each sum can occur. i.e. how many ways can we get a sum of 15?. Look at the coefficient of the x^15 term.
It is 6. There are 6 ways to get a sum of 15. How many ways can we get a sum of 20?. Look at x^20 and we see its coefficient is 1. There is 1 way to get a sum of 20, and that is by drawing two 10's.
Since there are 100 possible outcomes, count up the coefficients of the even exponents. They sum to 50. Therefore, the probability is of the two balls summing to an even number. The other half sums to odd numbers.

Answer by Alan3354(69443) (Show Source):
You can put this solution on YOUR website!
The 2nd ball must match the 1st in "oddness or evenness"
If the 1st ball chosen in odd, there are 4 remaining odds out of 9.
Same if it's even.
--> 4/9
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PS It's not possible to get a sum of 20, as the other tutor mentions.