Question

From ”The Basic Practice of Statistics” by Moore. Many people who come to clinics to be tested for HIV don’t come back to learn the test results. Clinics now use ”rapid HIV tests” that give a result while the client waits. In a particular clinic use of rapid tests increased the percent of clients who learned their test results from 69% to 99.7%.

The trade-off for fast results is that rapid tests are less accurate than slower laboratory tests. Applied to people who have no HIV antibodies, one rapid test has probability 0.004 of producing a false positive.

• If a clinic tests 200 people who are free of HIV antibodies, what is the chance that no false positives occur?

- What is the probability that at least one false positive occurs?

Answer 1

Solution

Back-up Theory

If X ~ B(n, p). i.e., X has Binomial Distribution with parameters n and p, where n = number of trials and

p = probability of one success, then, probability mass function (pmf) of X is given by

p(x) = P(X = x) = (ⁿC_x)(p^x)(1 - p)^{n – x}, x = 0, 1, 2, ……. , n ………….........................................................................………..(1)

[This probability can also be directly obtained using Excel Function: Statistical, BINOMDIST].............................………….(1a)

P(at least one) = 1 – P(none) ……………………………………………………………….................................................……..(2)

Now to work out the solution,

Let

X = Number of cases out of 200 tested by the clinic which shows false positive. Then,

X ~ B(200, 0.004), where 0.004 = probability of producing a false positive ..........................................................................(3)  

Part (a)

The chance that no false positives occur

= P(X = 0)

= (²⁰⁰C₀)(0.004⁰)(0.996)²⁰⁰ [vide (1) and (3)]

= 0.4486 [vide (1a)]   Answer 1

Part (b)

The probability that at least one false positive occurs

= 1 - P(X = 0) [vide (2)]

= 1 – 0.4486 [vide Answer 1]

= 0.5514 Answer 2

DONE