TITLE: Risk of AIDS Infection - Overestimation?
QUESTION [7 upvotes]: As an example of a probability misunderstanding, the book "Mathematics Statistics and Data Analysis" by John A. Rice gives the following quote from the Los Angeles Times:
"Several studies of sexual partners of people infected with the virus show that a single act of unprotected vaginal intercourse has a surprisingly low risk of infecting the uninfected partner--perhaps one in 100 to one in 1,000. For an average, consider the risk to be one in 500. If there are 100 acts of intercourse with an infected partner, the odds of infection increase to one in five.
Statistically, 500 acts of intercourse with one infected partner or 100 acts with five different infected partners lead to a 100% probability of infection (statistically, not necessarily in reality)." 
The full article is here.
Rice explains that this is flawed by considering just two acts of intercourse: if we let $A_1$ denote the event that infection occurs on the first act and $A_2$ the event infection occurs on the second, then the event that infection occurs is $B = A_1\cup A_2$ and
$$P(B) = P(A_1) + P(A_2) - P(A_1\cap A_2) \leq P(A_1) + P(A_2) = {2 \over 500}$$
But I'm still confused: I understand that the above is trying to show that the article is overestimating, but surely the probability of $A_1 \cap A_2$ is $0$? You can only be infected with AIDS once... as far as I know. And even if it is not $0$, what's stopping us from exceeding the $P = 1$ threshold eventually if we take enough events (which clearly seems nonsensical since its intuitively its clear that someone might avoid AIDS indefinitely)?
Just to clarify, I'm not disagreeing that the article is flawed, I'm just struggling to locate the exact flaw. Many thanks.

REPLY [3 votes]: I'm not disagreeing that the article is flawed, I'm just struggling to locate the exact flaw.

The exact flaw is that the article thinks probabilities are additive. In reality they are not (unless the events are mutually exclusive).
For example, if you flip a coin, the probability of heads is 50%. The article would tell you that if you flip the coin twice the probability of getting heads at least once is 100%. In reality there's a 25% chance of getting 2 heads, 50% chance of getting 1 heads and 1 tails, and 25% chance of getting 0 heads. So the probability of at least one heads is 75%.