# A virus has been spread around a population. The prevalence of this virus is 84%. A diagnostic test, with a specificity of 94% and sensitivity of 15%, has been introduced. If a patient is drawn randomly from the population, what is the probability that: a) a person has the virus, given that they tested positive? b) a person has the virus, given that they tested negative?

A virus has been spread around a population. The prevalence of this virus is 84%. A diagnostic test, with a specificity of 94% and sensitivity of 15%, has been introduced. If a patient is drawn randomly from the population, what is the probability that:
a) a person has the virus, given that they tested positive?
b) a person has the virus, given that they tested negative?
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Liam Everett
First of all Let's define what Sensitivity and Specificity of a test are:
- Sensitivity is defined as
$\mathbb{P}\left[{T}^{+}|D\right]$
- Specificity is defined as
$\mathbb{P}\left[{T}^{-}|\overline{D}\right]$
Where ${T}^{+},{T}^{-}$ indicate positive and negative test result while $D$ is "disease"
Second let's take 10,000 persons and see what is happening with the given probabilities

What you are requested to calculate is
(a) $\mathbb{P}\left[D|{T}^{+}\right]=\frac{1260}{1356}\approx 92.92\mathrm{%}$
and
(b) $\mathbb{P}\left[D|{T}^{-}\right]=\frac{7140}{8644}\approx 82.60\mathrm{%}$