How to Write Mathematics

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:

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:

(1) $\displaystyle P(A | B) = \frac{P(A) \times P(B | A)}{P(B)}$

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 equation (1), $P(A | B) \approx 0.32$ . 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%.

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Page last modified 25 October 2012 at 06:14 PM