Math Problem Statement

A certain virus infects one in every 200 people. A test used to detect the virus in a person is positive 90% of the time if the person has the virus and 8% of the time if the person does not have the virus. (This 8% result is called a false positive.) Let A be the event "the person is infected" and B be the event "the person tests positive".

a) Find the probability that a person has the virus given that they have tested positive, i.e. find P(A|B). Round your answer to a percent rounded to the nearest fourth decimal place and do not include a percent sign. P(A|B)= %

b) Find the probability that a person does not have the virus given that they test negative, i.e. find P(A'|B'). Round your answer as a percent rounded to the nearest fourth decimal place and do not include a percent sign. P(Ā|B̄) = %

Solution

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Math Problem Analysis

Mathematical Concepts

Probability
Conditional Probability
Bayes' Theorem
Law of Total Probability

Formulas

Bayes' Theorem: P(A|B) = [P(B|A) * P(A)] / P(B)
Law of Total Probability: P(B) = P(B|A) * P(A) + P(B|A') * P(A')

Theorems

Bayes' Theorem
Law of Total Probability

Suitable Grade Level

Grades 11-12