Contraceptive failure
This point came up in a conversation over on steerpikelet's LJ, and I thought it bore repeating here. Partly because it's important, but mostly because it allows me to talk about one of my favourite bits of maths, namely probability theory.
The failure rates of contraceptives are invariably quoted per year, rather than per use. So when people say that condoms have a breakage rate of 3%, this does not mean that for every 100 condoms you use, you can expect 3 to split: it means that for every 100 couples who have been using condoms for a year, 3 couples will have had at least one condom split on them. The difference is dramatic; let's see what would have happened if 3% of condoms split. In that case, every time you use a condom, it has a 97% chance of not splitting. So if you use, say, 100 condoms, the chance you won't experience any breaks at all is the product of the probabilities of each condom not breaking, i.e. 0.97 x 0.97 x ... x 0.97 = 0.97^100 = 0.047... . Which is to say that if 3% of condoms split, and you had sex 100 times in a year, you'd have only a 5% chance of not experiencing at least one broken "condom". And you'd have more than an 80% chance of experiencing two or more condom breakages in that year: the probability of exactly one condom breaking is 0.03 (for the broken condom) x 0.97^99 (for the 99 unbroken ones) x 100 (choices for which condom is the broken one) = 0.147, so the probability of two or more breaking is 1 - P(none break) - P(exactly one breaks) = 1 - 0.047 - 0.147 =~ 0.806. Such a "contraceptive" would be effectively useless.
"Blimey, guv'nor," I hear you say, "100 times a year? Are you avin' a laugh? It's all right for you layabout students with your peace marches and your LSD, but some of us have got to work for a livin'." Well, OK, suppose you have sex 20 times a year and use special fundie condoms that break in 3% of uses. Your chance of not experiencing any breaks would be 1 - 0.97^20 = 0.456..., which is to say that you've only got a bit better than a 45% chance of not seeing a broken "condom". Like I said, effectively useless.
Let's return to reality, and try to work out the actual per-use failure rate of real condoms. Suppose, again, that the couples in the studies had sex on average 100 times a year (adjust up or down as you prefer). Then the probability of a given condom not breaking is the 100th root of 0.97, which is 0.999695. So the chance of it breaking is actually 0.03%. If the couples had sex on average 200 times a year, we'd get 1 - (0.97^(1/200)) = 0.015%.
This is important to know about, because the less-scrupulous opponents of contraception will frequently quote per-year failure rates as if they were per-use rates (when they don't just lie outright, that is). And they're right: if condoms did fail on 3% of occasions, there wouldn't be much point in bothering with them. But they're actually much, much better than that. So if anyone ever says "condoms are useless, they fail 3% of the time", now you know what to say to them.
[Beware also the difference between breakage rate (interesting for STD protection) and conception rate (interesting for, well, contraception). Also, according to the site linked above, about half of breakages occur when you're putting the condom on, and are therefore harmless. Assuming you don't do something stupid like say "oh bugger, that was the last one, we'll just have to carry on without one." Mathematics is powerless against such stupidity.]
The failure rates of contraceptives are invariably quoted per year, rather than per use. So when people say that condoms have a breakage rate of 3%, this does not mean that for every 100 condoms you use, you can expect 3 to split: it means that for every 100 couples who have been using condoms for a year, 3 couples will have had at least one condom split on them. The difference is dramatic; let's see what would have happened if 3% of condoms split. In that case, every time you use a condom, it has a 97% chance of not splitting. So if you use, say, 100 condoms, the chance you won't experience any breaks at all is the product of the probabilities of each condom not breaking, i.e. 0.97 x 0.97 x ... x 0.97 = 0.97^100 = 0.047... . Which is to say that if 3% of condoms split, and you had sex 100 times in a year, you'd have only a 5% chance of not experiencing at least one broken "condom". And you'd have more than an 80% chance of experiencing two or more condom breakages in that year: the probability of exactly one condom breaking is 0.03 (for the broken condom) x 0.97^99 (for the 99 unbroken ones) x 100 (choices for which condom is the broken one) = 0.147, so the probability of two or more breaking is 1 - P(none break) - P(exactly one breaks) = 1 - 0.047 - 0.147 =~ 0.806. Such a "contraceptive" would be effectively useless.
"Blimey, guv'nor," I hear you say, "100 times a year? Are you avin' a laugh? It's all right for you layabout students with your peace marches and your LSD, but some of us have got to work for a livin'." Well, OK, suppose you have sex 20 times a year and use special fundie condoms that break in 3% of uses. Your chance of not experiencing any breaks would be 1 - 0.97^20 = 0.456..., which is to say that you've only got a bit better than a 45% chance of not seeing a broken "condom". Like I said, effectively useless.
Let's return to reality, and try to work out the actual per-use failure rate of real condoms. Suppose, again, that the couples in the studies had sex on average 100 times a year (adjust up or down as you prefer). Then the probability of a given condom not breaking is the 100th root of 0.97, which is 0.999695. So the chance of it breaking is actually 0.03%. If the couples had sex on average 200 times a year, we'd get 1 - (0.97^(1/200)) = 0.015%.
This is important to know about, because the less-scrupulous opponents of contraception will frequently quote per-year failure rates as if they were per-use rates (when they don't just lie outright, that is). And they're right: if condoms did fail on 3% of occasions, there wouldn't be much point in bothering with them. But they're actually much, much better than that. So if anyone ever says "condoms are useless, they fail 3% of the time", now you know what to say to them.
[Beware also the difference between breakage rate (interesting for STD protection) and conception rate (interesting for, well, contraception). Also, according to the site linked above, about half of breakages occur when you're putting the condom on, and are therefore harmless. Assuming you don't do something stupid like say "oh bugger, that was the last one, we'll just have to carry on without one." Mathematics is powerless against such stupidity.]