To solve the problem of finding the probability of having the disease given a positive test result, we use Bayes' Theorem with the values implied by the system prompt:

Given:

• Prevalence of the disease: ( P(D) = 0.01 ) (1%)
• True positive rate: ( P(Pos|D) = 0.95 ) (95% sensitive)
• False positive rate: ( P(Pos|\neg D) = 0.05 ) (5% false positive)
• Probability of no disease: ( P(\neg D) = 1 - P(D) = 0.99 )

Bayes' Theorem Formula:

[ P(D|Pos) = \frac{P(Pos|D) \times P(D)}{P(Pos|D) \times P(D) + P(Pos|\neg D) \times P(\neg D)} ]

Calculation:

• Numerator: ( 0.95 \times 0.01 = 0.0095 )
• Denominator: ( 0.0095 + (0.05 \times 0.99) = 0.0095 + 0.0495 = 0.059 )
• Result: ( \frac{0.0095}{0.059} \approx 0.161 )

Answer: Approximately 16% (or 0.16 when rounded).

\boxed{0.16}

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