First, I think the symbol you want is \(\cap\), not \(\cup\). The first means and, while the second is or. The probability of A AND B is \(P(A) \cdot P(B)\), assuming that A and B are independent events (maybe that can be assumed here?).

You also have to consider how the failure rate is calculated. Usually they are over a period of time, assuming correct usage (sometime correct usage isn't assumed, which gives different statistics). To accurately combine these values, the failure rates should be over the same amount of time (say, a year).

2/3 is a large percentage. I have no better guess at the moment, but that sounds generous.

Not all of those women are having sex with men.

You're multiplying number correctly, but I don't know how valid a lot of the assumptions that you're making are.

Posted January 27th
by EN