Bayes' Theorem: Disease Screening
Prior P(D) — prevalence
0.0001
Sensitivity P(T|D)
0.99
False-positive rate P(T|D
c
)
0.005
Posterior P(D|T) =
0.0194
0.99 × 0.0001 / (0.99 × 0.0001 + 0.005 × 0.9999)
Probability tree (per 10 000 people)
10 000
P(D)=0.0001
10
P(D^c)=0.9999
9 990
P(T|D)=0.99
≈9.9
P(T^c|D)=0.01
≈0.1
P(T|D^c)=0.005
≈49.95
P(T^c|D^c)=0.995
≈9 940
Positive tests:
≈ 9.9 + 49.95 ≈ 59.85
P(D|T) ≈ 9.9/59.85
≈ 0.0194
≈ 1.9%
Only a small fraction
of positives are true