[samtheman]

Using your assumptions: condom rate = 3% (which I doubt) and infected-partner-possibility = 10%:

Chances of a single act of sex being "unlucky" =

(the chance of condom failing) * (the chance of the partner being infected) =

.03 * .10 = .003.

Next: the chance of a single act of sex being "lucky" =

1 - "unlucky chance" = .997.

Next: chance of sex remaining "lucky" through 10 encounters =

.997 raised to the 10th power

= .997 ^ 10 = .970402

and the chance of getting infected after 10 encounters =
1 - .970402 = .029598

Next: the chance of remaining lucky through 100 encounters =

.997 raised to the 100th power

= .997 ^ 100 = 0.74048426

and the chance of being infected after 100 encounters =
1 - 0.74048426 = 0.74048426

Next: the chance of remaining lucky through 1000 encounters =

.997 raised to the 1000th power

= .997 ^ 1000 = 0.049563083


and the chance of being infected after 1000 encounters =
1 - 0.049563083 = 0.950436917


So chances are, under those conditions, 1000 encounters means 95% of contagion.

I would argue that your premise is false. I think 3% failure rate of condoms is a very high figure.

[NukeMan]

I think 3% failure rate is actually a decent estimate, and in fact *overestimates* their effectiveness, but I am too lazy to Google the actual stats. The infected-partner possibility is 20% (not 10%) according to the posters estimate.

There is one additional factor to be included. IIRC, the CDC estimated that only 1 in 500 sex acts (with a partner you knew *for sure* was infected) led to you catching aids. Now, this is a general number combining all genders and sex acts. Many sex acts are riskier than others - receptive anal intercourse is the riskiest (much greater than 1 in 500). So, taking into account all that:

Chance of infection per episode assuming 3% failure, 20% chance of partner infection, and 1/500 infection rate is .000012. I would here disagree with the 3% failure rate as being too low, though.

[Ronaldus Magnus]

and the chance of being infected after 100 encounters = 1 - 0.74048426 = 0.74048426

1 - .740 = .260

[Maelstrom]

Condom failure is much higher than 3%.

The 3% figure only accounts for the probability of a condom to fail due to manufactured faults in condoms.