10% of the population in a town is HIV+. A new diagnostic kit for HIV detection is available; this kit correctly identifies HIV+ individuals 95% of the time, and HIV- individuals 90% of the time. A particular patient is tested using this kit and is found to be positive. The probability that the individual is actually positive is ___.

GATE 2014 · General Aptitude · Conditional Probability · medium

Answer: P(HIV+ | test+) ≈ 0.4896 (NAT answer range: 0.4895 to 0.4897)

1. Identify the false positive rate: P(test+ | HIV-) = 1 - 0.90 = 0.10
2. Compute total probability of testing positive: P(test+) = (0.95)(0.10) + (0.10)(0.90) = 0.095 + 0.090 = 0.185
3. Apply Bayes' theorem: P(HIV+ | test+) = 0.095 / 0.185 = 95/185 = 19/37 ≈ 0.5135