How to Write MathematicsPressBooks.HowToWriteMathematics HistoryHide minor edits - Show changes to output 25 October 2012 at 06:14 PM
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Suppose you are deciding whether to approve or decline a visa application. You run a test to assess whether the applicant presents a risk. Let us suppose that the test is accurate 90% of the time (regardless of whether the result is positive or negative). That is, 10% of the time the test returns a false negative or a false positive. You know that visa applications are declined rarely and about 95% are approved. If your test result comes back positive (the applicant presents a risk), what are the chances that the applicant will be declined on investigation? Do you think it is approximately: #%alpha% 0.9 # 0.7 # 0.5 # 0.3 # 0.1 To work out the answer, we use Bayes' theorem, which tells us how to find the probability of event A given event B, written {$P(A | B)$}, in terms of the probability of B given A, written {$P(B | A)$}, and the probabilities of A and B: =|{$bayes, \displaystyle P(A | B) = \frac{P(A) \times P(B | A)}{P(B)} $} In this case, event A is the event that you decline the application and event B is the event that the test is positive (the applicant presents a risk). In our example, {$P(A) = 0.05$} and {$P(B | A) = 0.9$}. We can derive {$P(B)$} -- the probability of a positive test -- by adding the probabilities of a true positive test on a decline and a false positive test on an approval. {$* P(B) = P(B | A) \times P(A) + P(B | \text{not }A) \times P(\text{not }A) $} {$ = 0.9 \times 0.05 + 0.1 \times 0.95 $} This gives {$P(B) = 0.14$} and so from EQ(bayes), {$P(A | B) \approx 0.32$}. In other words, the correct answer is (d) -- you are more than twice as likely to see a false positive than a true positive. We can also calculate the probability of a false negative. That is, the risk assessment test returns a negative result, indicating that applicant does not present a risk, even though the applicant actually does. Applying Bayes theorem, {$P(A | \text{not }B) \approx 0.0058$} -- about 0.6%. |