Final answer:
To determine the probability that a person has the virus given they've tested positive, Bayes' theorem is used with the given true positive and false positive rates. The result is a probability of approximately 5.2%, which means that if someone tests positive, there's about a 5.2% chance they actually have the virus.
Step-by-step explanation:
The question pertains to the field of probability and conditional probability, where we aim to find the probability that a person has a virus given that they have tested positive for it. In this particular scenario, we know that 0.8% of the population is affected by the virus. The test has an 88% true positive rate and a 13% false positive rate.
To solve for the probability that a person has the virus given a positive test result, we use Bayes' theorem which relates the conditional and marginal probabilities of stochastic events.
First, let's set up the variables:
- P(Virus) = Probability of having the virus = 0.8% or 0.008
- P(No Virus) = Probability of not having the virus = 1 - P(Virus) = 0.992
- P(Positive|Virus) = Probability of testing positive given having the virus = 88% or 0.88
- P(Positive|No Virus) = Probability of testing positive given not having the virus = 13% or 0.13
Using Bayes' theorem:
P(Virus|Positive) = (P(Positive|Virus) × P(Virus)) / (P(Positive|Virus) × P(Virus) + P(Positive|No Virus) × P(No Virus))
Calculated numbers:
P(Virus|Positive) = (0.88 × 0.008) / ((0.88 × 0.008) + (0.13 × 0.992))
P(Virus|Positive) = 0.00704 / (0.00704 + 0.12896)
P(Virus|Positive) = 0.00704 / 0.136
P(Virus|Positive) = 0.05176470588235, or approximately 5.2%
Therefore, the probability that a person has the virus given that they have tested positive is about 5.2%, when rounded to the nearest tenth of a percent.