Probability with a diagnostics tests

Question

Question: 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?

(As a follow up) Will the positive results in this test be mostly false positives?

Attempts and Ideas: I'm nearly certain that Bayes' Theorem.

$$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$

although I am unsure how the information given relates to this theorem. Perhaps $P(A)$ is testing positive and $P(B)$ is actually being positive?

Any help or guidance is greatly appreciated!

Answer 1

First of all Let's define what Sensitivity and Specificity of a test are:

• Sensitivity is defined as

$$\mathbb{P}[T^+|D]$$

• Specificity is defined as

$$\mathbb{P}[T^-|\overline{D}]$$

Where $T^+,T^-$ indicate positive and negative test result while $D$ is "disease"

Second let's take (as an example) 10,000 persons and see what is happening with the given probabilities

What you are requested to calculate is

(a)

$$\mathbb{P}[D|T^+]=\frac{1260}{1356}\approx 92.92\%$$

and

(b)

$$\mathbb{P}[D|T^-]=\frac{7140}{8644}\approx 82.60\%$$