What is the probability of contracting HIV from multiple encounters?

By MUNGAI KIHANYA

The Sunday Nation

Nairobi,

02 March 2008

Back in 2005, I wrote about the probabilities of contracting HIV through sexual intercourse. From the calculations, it turned out that every day (night!) in that year, 35,000 HIV-negative Kenyans engaged in unprotected sex with HIV-positive partners. Now, even I thought that number was too high, but not any more – after receiving this email from a reader who conceals his name.

He says that he had approximately 40 unprotected sexual encounters with a certain lady only to discover later that she is HIV-positive. He writes: “I read somewhere that the chances of contracting the virus from of unprotected sex from female to male is 0.2%. What are my percentage chances of having contracted the virus?”

This is a good example of how NOT to use mathematics. Firstly, why estimate the value of a quantity that can be measured directly and easily? For this reason, I advised the reader to go and get tested without delay (as I told him, no amount of mathematical manipulations will change the outcome!) Secondly, probabilities only work when there are large populations and, therefore, they are only useful for planning purposes.

For example, assuming that the 0.2 percent probability of contraction is correct, then if 100,000 men had unprotected sex with HIV-positive partners, then we can expect that only about 200 of them would catch the virus. However, it is not possible to predict which of the men will be infected. From the point of view of each one of them, the only possible outcomes are either infected or not infected – in other words, 50-50!

Nonetheless, suppose that after the first experiment, we remove the 20 infected men and repeat it with 99,800. The result would almost 200 infected (199.6, but you can’t have a 0.6 of a man!). If we repeated the experiment again, we would have another 199 infections. Now, suppose we repeat this procedure 40 times; how many HIV-NEGATIVE men are likely to have remaining?

To find out, we multiply the probability of not being infected by itself 40 time (that is, 0.998 x 0.998 x 0.998 x … x 0.998, 40 times) and then multiply the result by 100,000 – the original number. The answer is 92,304, meaning that a total of 7,696 will have been infected. Thus we can conclude that the probability of infection after 40 encounters is about 7.7percent.

But all this is of no use to our reader: It only helps the planners in the health ministry – they can use the information in deciding what quantity of, say, ARVs are required in a year.