So, given that person A has HIV, the probability that person C contracts it on any one time is 50/10000= 0.005.Every 10,000 times Person A had sex with Person C, 50 of the 10,000 intercourse episodes would result in Person C contracting HIV (if person B already gave Person A HIV).
Given that person A has HIV, the probability person C does NOT contract HIV each time is 1- 0.005= 0.995. The probability person C does NOT contract HIV in two times is (0.995)2= 0.990025 so the probability person C does contract HIV is 1- .990025= 0.009975.What is the percent chance that Person B gave Person A HIV *AND* Person A subsequently gave Person C HIV? It is understood that Person A and Person B engaged in intercourse five times and then after that Person A and Person C engaged in intercourse twice.
The probability that person A contracts HIV and that person C then contracts it from person A is (0.00032496)(0.009975)= 0.0000032415.