I do not know the chances of getting a false positive or false negative as far as percents go or anything like that.
There are a few different causes for false negatives and false positives.
You might get a false negative if you test too soon, before the anitbodies are made. That is the most common reason for getting a false negative.
False positives often happen for no reason other than that the tests are designed to be sensitive. It's better to get a false positive than a false negative. It might be stressful, but it makes the doctors take another look and do another test. A false negative would be dangerous because people might not take that second look. So tests are often designed to give false positives easier than false negatives.
Other times you might get the wrong result just because someone did not do the test right.