Question

The prevalence of flu in a population is 1%. A diagnostic test has a false positive rate of 10% and a false negative rate of 10%. The probability that a randomly chosen person tests positive is ________ (Round off to three decimal places).

Hint:

Bayes' Theorem helps in calculating conditional probabilities like this one. Remember to use the prevalence and false positive/negative rates correctly in the formula.

Answer 1

Correct Answer: 0.107

We can use Bayes' Theorem to solve this problem. Let:
- \( P(\text{Flu}) = 0.01 \) (prevalence of flu),
- \( P(\text{No Flu}) = 0.99 \) (probability of not having the flu),
- \( P(\text{Positive} \mid \text{Flu}) = 0.9 \) (probability of testing positive given flu, i.e., 1 - false negative rate),
- \( P(\text{Positive} \mid \text{No Flu}) = 0.1 \) (false positive rate).
The probability of testing positive, \( P(\text{Positive}) \), is: \[ P(\text{Positive}) = P(\text{Positive} \mid \text{Flu}) P(\text{Flu}) + P(\text{Positive} \mid \text{No Flu}) P(\text{No Flu}). \] Substituting the known values: \[ P(\text{Positive}) = (0.9)(0.01) + (0.1)(0.99) = 0.009 + 0.099 = 0.108. \] Thus, the probability that a randomly chosen person tests positive is: \[ \boxed{0.108}. \]