They're not really wrong more times than they're rights, as they don't give a "yes" or "no" answer, so saying that they give false negatives or false positives is misleading (in the way we would understand that).
The tests you can choose (or not) are screening tests which give the likelihood, based on a range of factors, that your baby has down syndrome.
The result will be a ratio, for eg 1:2000. That means that if you took 2000 pregnancies with the same result as yours, one of those babies would have DS. 1999 would not.
A higher risk result is generally taken to be around 1:150 at which point a diagnostic test might be offered which would give a definite yes/no answer (i say definite, there are very, very rare situations where this is not the case).
Even if you have a higher risk result there's more chance than not that your baby is unaffected. You can choose not to have a diagnostic test (there is a risk of miscarriage).
There is also now the option of a much more precise screening test called Harmony or Nifty. This is a blood test and carries no risk to the pregnancy. I think it's available on the NHS in some places although you may still have to go private.
You should think about what you would do if you got a higher risk result. If you feel that you would refuse the diagnostic test and carry on with the pregnancy no matter what then you might consider missing out on the screening tests.
Ultimately it's your choice to decide what's best for you.