Is it scientifically possible to determine what sex a certain male would produce

For example, could I get my balls checked to see if I have boys and girls down there?

Half your sperm contain X chromosomes, and half your sperm contain Y chromosomes. The split is exactly 50/50. How do I know? Because you are male, every cell in your body (except for the sperm) contains an X chromosome paired with a Y choromosome. When your sperm are formed, they form from a cell that divides and splits up the XY pair, and the X goes into one sperm, and the Y goes into another sperm. Every time this happens, you get one X sperm and one Y sperm, so you always have an equal amount of the two.

Every egg contains an X chromosome. If a Y sperm wins the race to the egg, you get XY, a male. If an X sperm wins the race, you get XX, which is female.

Not really. Its not as though you're producing only X or Y sperm. Both are produced, but only one fertilizes the egg. That said, certain men ARE more likely to fertilize with a certain gender due to the health of certain genders of sperm. For some, one will be far stronger and more likely to reach the egg than the other gender.
EDIT: Men do determine gender. Females only contribute and X, so its the male genetic contribution that determines gender.

You have both of them. Millions of them. Boys are Y chromosomes while girls are X chromosomes. A female's egg always has an X chromosome. So, if your X chromosome meets with her X chromosome, you have XX which makes a girl. If you're Y chromosome meets her X chromosome, you have XY or YX which makes a boy. Your sperm cells literally fight and kill each other to make it to the egg. Determining which one makes it is not possible, but there are ways to increase your X or Y chromosome counts. Google it, I'm sure there are old remedies and pills out there that make it happen. The only down side is genetic defects

The problem is, that every given sample of sperm existing in the man's system at any given instant in time, is only going to live there for a few days.

Several months later, you won't have the same sperm. It will be a completely different composition, and have a completely different preliminary sex ratio.

Lol you're an idiot its not just men who determine sex.its also.women.and.its.random you dont just produce boys or girls you will have.almost an equal chance of produxing.each

No you cannot!