What's wrong with my solution to this probability problem Let $S$ be the set of all integers between −10 and 10, inclusive: $\{−10, −9, −8, . . . , 8, 9, 10\}$. If two integers are chosen out of the set at random, then the probability that the product of the two integers is positive is a/b, where a and b share no common factors. Find a + b. 
I found that the probability that the product is positive is $200/441$, and the probability that the product is negative is $200/441$. However, I found that the probability the product is zero as $1/21$, since only 0 has to be the first factor. This leaves us with $20/441$ extra, meaning that something is wrong. However, I don't know what's wrong! Can someone help?

The product is $0$ if either sample is $0$.  $\frac{1}{21}$ is the chance that the first sample is zero. 
Assuming the samples are uniform and independent, the probability of a 0 product is 
$$Pr(x_0 = 0 {\rm \ or \ } x_1=0) \\
= Pr(x_0 = 0) + Pr(x_1=0) - Pr(x_0 = 0 {\rm \ and \ } x_1 = 0) \\
= \frac{1}{21} + \frac{1}{21} - \frac{1}{21^2} \\
= \frac{41}{441}$$
The positive is $\frac{200}{441}$ and negative is $\frac{200}{441}$ 
so the sum of probabilities of positive negative and $0$ products are
$$ \frac{200}{441} + \frac{200}{441} + \frac{41}{441} = 1$$
They are mutually exclusive and cover all outcomes.
