[jointpdf]

via Bayes Rule: “Assuming an underlying infection rate P(I), what’s the probability that a person is actually immune (=was infected), given that they test positive, i.e. P(I|+)?”:

  P(I|+) 
  = 
  P(+|I)*P(I) / P(+)
  = 
  Sens*P(I) / [Sens*P(I) + (1-Spec)*(1-P(I))]
  =
  .01 / (.01 + .001*.99)

This is exhibit A of the base rate fallacy (https://en.m.wikipedia.org/wiki/Base_rate_fallacy).