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Chapter 7: Problem 23

If a certain disease is present, then a blood test will reveal it \(95 \%\) of the time. But the test will also indicate the presence of the disease \(2 \%\) of the time when in fact the person tested is free of that disease; that is, the test gives a false positive \(2 \%\) of the time. If \(0.3 \%\) of the general population actually has the disease, what is the probability that a person chosen at random from the population has the disease given that he or she tested positive?

Short Answer

Expert verified
The probability that a person chosen at random from the population has the disease given that he or she tested positive is approximately 9.85%.

Step by step solution

01

Understand the problem and notations

Let's assign some notations to the given information: - A: event that a person has the disease. - B: event that a person tests positive for the disease. We want to find the conditional probability P(A|B) - that is, the probability that a person has the disease given that he or she tested positive.
02

Apply Bayes' Theorem

Bayes' theorem relates the conditional probabilities P(A|B) and P(B|A). It is given by the following formula: \(P(A|B) = \frac{P(B|A) * P(A)}{P(B)}\) Here, we have P(B|A) which is 0.95, the probability of a person testing positive given that they have the disease, and P(A) which is 0.003, the probability of a person having the disease.
03

Find P(B)

To find P(B), the probability that a person tests positive regardless of whether they have the disease or not, we can use the Law of Total Probability: \(P(B) = P(B|A) * P(A) + P(B|\overline{A}) * P(\overline{A})\) Here, \(\overline{A}\) represents the event that a person does not have the disease. We are given P(B|\(\overline{A}\)) = 0.02, which is probability of a false positive.
04

Calculate P(\(\overline{A}\))

Since A is the event that a person has the disease and P(A)=0.003, the probability of a person not having the disease is: \(P(\overline{A}) = 1 - P(A) = 1 - 0.003 = 0.997\)
05

Calculate P(B)

Using the values obtained, we can now calculate P(B): \(P(B) = P(B|A) * P(A) + P(B|\overline{A}) * P(\overline{A}) = 0.95 * 0.003 + 0.02 * 0.997 = 0.02885 \)
06

Calculate P(A|B)

Now that we have P(B), we can use Bayes' theorem to find P(A|B): \(P(A|B) = \frac{P(B|A) * P(A)}{P(B)} = \frac{0.95 * 0.003}{0.02885} = 0.09847 \) Therefore, the probability that a person chosen at random from the population has the disease given that he or she tested positive is approximately 9.85%.

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Most popular questions from this chapter

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. Let \(S=\left\\{s_{1}, s_{2}, \ldots, s_{n}\right\\}\) be a uniform sample space for an experiment. If \(n \geq 5\) and \(E=\left\\{s_{1}, s_{2}, s_{5}\right\\}\), then \(P(E)=3 / n\).

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