Question

Rapid HIV tests allow for quick diagnosis without expensive laboratory equipment. However, their efficacy has been called into question. In a population of 1517 tested individuals in Uganda, 4 had HIV but tested negative (false negatives), 166 had HIV and tested positive, 129 did not have HIV but tested positive (false positives), and 1218 did not have HIV and tested negative. A randomly slected person from this population tests negative for HIV. What is the probability that this person has HIV?

Answer 1

The probability that a person has HIV given that the test negative is 0.0033.

Explanation:

Denote the events as follows:

X = a person in Uganda had HIV

Y = a person in Uganda was tested positive for HIV.

The information provided is:

`P(Y^(c)\cap X)=(4)/(1517)\\`

`P(Y\cap X)=(166)/(1517)\\`

`P(Y\cap X^(c))=(129)/(1517)\\`

`P(Y^(c)\cap X^(c))=(1218)/(1517)\\`

The probability that a person has HIV given that he/she was tested negative is:

`P(X|Y^(c))=(P(Y^(c)\cap X))/(P(Y^(c)))`

Compute the probability of a person not having HIV as follows:

`P(Y^(c))=P(Y^(c)\cap X)+P(Y^(c)\cap X^(c))=(4)/(1517)+(1218)/(1517) =(4+1218)/(1517) =(1222)/(1517)`

Compute the value of
`P(X|Y^(c))` as follows:

`P(X|Y^(c))=(P(Y^(c)\cap X))/(P(Y^(c)))=(4)/(1517)*(1517)/(1222)=0.0033`

Thus, the probability that a person has HIV given that he/she was tested negative is 0.0033.