Question

Swaziland has the highest HIV prevalence in the world : 25.9% of this country’s population is infected with HIV. The ELISA test is one of the first and most accurate tests for HIV. For those who carry HIV, the ELISA test is 99.7% accurate. For those who do not carry HIV, the ELISA test is 92.6% accurate. 1. If an individual from Swaziland has tested positive, what is the probability that he carries HIV ? 2. If an individual from Swaziland has tested negative, what is the probability that he is HIV free ?

Answer 1

1. If an individual from Swaziland has tested positive, what is the probability that he carries HIV ?

P=0.8249 or 82.49%

2. If an individual from Swaziland has tested negative, what is the probability that he is HIV free ?

P=0.9988 or 99.88%

Explanation:

Make the conditional probability table:

Individual

Infected Not infected

ELISA

Positive

Negative

Totals

The probability of an infected individual with a positive result from the ELISA is obtained from multiplying the probability of being infected (25.9%) with the probability of getting a positive result in the test if is infected (99.7%), the value goes in the first row and column:

P=0.259*0.997=0.2582 or 25.82%

Individual

Infected Not infected Totals

ELISA

Positive 25.82%

Negative

Totals

The probability of a not infected individual with a negative result from the ELISA is obtained from multiplying the probability of not being infected (100%-25.9%=74.1%) with the probability of getting a negative result in the test if isn't infected (92.6%), the value goes in the second row and column:

P=0.741*0.926=0.6862 or 68.62%

Individual

Infected Not infected Totals

ELISA

Positive 25.82%

Negative 68.62%

Totals

In the third row goes the total of the population that is infected (25.9%) and the total of the population free of the HIV (74.1%)

Individual:

Infected Not infected Totals

ELISA

Positive 25.82%

Negative 68.62%

Totals 25.9% 74.1%

Each column must add up to its total, so the probability missing in the first column is 25.9%-25.82%=0.08%, and the ones for the second column is 74.1%-68.62%=5.48%.

Individual

Infected Not infected Totals

ELISA

Positive 25.82% 5.48%

Negative 0.08 68.62%

Totals 25.9% 74.1%

Individual

The third column is filled with the totals of each row:

Infected Not infected Totals

ELISA

Positive 25.82% 5.48% 31.3%

Negative 0.08 68.62% 68.7%

Totals 25.9% 74.1% 100%

The probability A of tested positive is 31.3% and the probability B for tested positive and having the virus is 25.82%, this last has to be divided by the possibility of positive.

P(B/A)=0.2582/0.313=0.8249 or 82.49%

The probability C of tested negative is 68.7% and the probability D for tested negative and not having the virus is 68.62%, this last has to be divided by the possibility of negative.

P(D/C)=0.6862/0.687=0.9988 or 99.88%