OK, I'll come across as a complete nerd, but here goes. To follow up on djconnel's comment, you can separate your (subjective) beliefs from (objective) evidence.
Evidence: test result (whatever that is, positive or negative, a numerical value,...)
hypothesis 1: rider dopes (and hypothesis 2: rider doesn't dope)
your belief: probability that rider dopes
Whatever your belief is, you can revise it once you have seen the test result, and this is how.
Prob(doping given result) Prob(result if doping) Prob(doping)
---------------------- = ------------------ * -----------
Prob(no_doping given result) Prob(result if no_doping) Prob(no_doping)
(edit: this should be 3 ratios, the system collapses my neat formula; this link shows it better: http://www.stat.auckland.ac.nz/~curran/statschat/bayes.png
The part on the right is your initial belief, given as an odds. If you think, before seeing the test, that the rider has a 50% chance of being doped, that translates as even odds (50:50), or 1.
The part in the middle is the evidence. In this case, maybe the probability of having low reticulocytes when you do blood transfusions is fairly high (eg 50%), but very low if you do not blood dope (eg 0.0001%, or 1 in a million). This ratio would be 500000.
The part on the left is what you should believe after you have seen that particular test. Now the odds are overwhelmingly in favor of the hypothesis that the rider dopes (in this example the odds are 500000, so way past 99.99%).
The nice thing is that people can disagree on the rightmost part, their prior beliefs, while agreeing on the evidence, the middle part. If your initial belief ws a near certainty that the rider does not dope (only 1 chance in 1 million), then even the damning evidence above will move the post-test odds to 1:2, or a 33% chance of doping - so now you are on the fence.