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Bayes' theorem updates a prior with new evidence: posterior = prior × sensitivity / P(positive). When the prior is low, even a very accurate test yields many false positives, so the posterior stays surprisingly small.
Bayes' theorem
P(D|+) = P(D)·P(+|D) / [P(D)·P(+|D) + P(¬D)·P(+|¬D)]
True positives are 0.0099 and false positives 0.0495, so the posterior is only 16.67% despite the accurate test.
Because the healthy group is so much larger that its 5% false-positive rate produces more positives than the sick group does.