Odds, probability, birth control effectiveness...

My last university math class was a looooooooooooooooooooong time ago. How do you figure this?

I’m rounding the numbers so I can follow the math…

The effectiveness of the pill is 95%.
The effectiveness of condoms is 90%.
The effectiveness of the sponge is 85%.

If she is on the pill and uses a sponge, and he wears a condom, the effectiveness is ____ .

What if only two methods are used?

If you assume that the various forms of birth control work independently, then the probability that at least one of the two works is 1 - (1 - p[sub]1[/sub])(1 - p[sub]2[/sub]), where p[sub]1[/sub] is the probability that the first one works, and p[sub]2[/sub] is the probability that the second one works. If you don’t assume that they’re independent, then you need to know how they interact in order to answer the question.

In case this has a real-life application, keep in mind that those effectiveness rates are based on a year of typical use, whatever that is. So you are assuming each of the 2-3 methods is used every time over the course of that typical year.

Further to ultrafilter’s answer, bear in mind that independence is a pretty reasonable assumption for things like hypothetical coin tossing, but a very strong (i.e. bold and probably wrong) assumption when dealing with human behaviour.

Yep…my understanding is that those effectiveness rates are lower than they might otherwise be due largely to humans screwing (no pun intended) up. For example the woman forgets to take the pill or the guy uses the condom wrong. There is of course still some chance these methods will not work even with perfect use (condoms sometimes break for instance).

Honestly those numbers always scared me. At times when I have been in a steady relationship sex over 100 times a year was normal. Based on those numbers I should have gotten her pregnant several times over (even with perfect use) in the course of that year. Never happened that way though so either you (and I) need to better understand exactly what those numbers are telling us or I just got very lucky.

NO. Based on those numbers you had a 10% chance (assuming condoms only) per year.

The numbers are NOT odds per instance of intercourse. They’re odds per a year’s worth of intercourse at typical rates.

That’s not really an issue of independence, though. What I’m thinking of are interactions between the various birth control methods when they are used correctly. I have a hard time imagining how any two methods mentioned in the OP might interact, but the default assumption has to be one of dependence.

There are lots of ways in which they can interact. For example, if someone uses multiple methods of birth control, would he/she be more or less likely to use each method correctly, compared to someone who just uses one method? It could go either way. If you’re that worried about pregnancy, maybe you’re less likely (than average) to miss a dose of the pill. On the other hand, if you’re on the pill, you may be less careful about using a sponge or condom every single time.

Assuming everything is independent, calculate the probability of all three failing and subtract it from 1.

Use the same principle.

The calculation everybody is discussing won’t work even in the absence of this sort of thing, if my understanding of the “effectiveness” numbers is correct. I think they’re the probability of not getting pregnant in a year of using the method. Even with no birth control, that probability is not zero. Let’s say, for the sake of illustration, that the probability of pregnancy with no birth control is 50%. Then the “effectiveness” of a method that does absolutely nothing is 50%. So consider the following methods:

Lucky rabbit’s foot: 50% effective
Horseshoe on wall: 50% effective
Saying “hocus pocus”: 50% effective

So, applying the above reasoning, if we use rabbit’s foot, horseshoe, and hocus pocus together, the chance of pregnancy is 1/8 or 12.5%, and the effectiveness is a respectable 87.5%. This is, of course, nonsense. The same overestimation of effectiveness occurs when this reasoning is applied to genuine birth control methods.

Uncertain, you raise a good point about the base not being zero. The number I’ve typically seen is 80% chance of pregnancy in a year with no birth control.

However, I don’t think your example completely holds. All three of your “methods” act through the same channel, luck. It’s somewhat parallel to taking three types of birth control pill. They can’t be assumed to act through separate channels. The examples in the OP can be assumed to have a cumulative effect, because they do act through separate channels.

There is no doubt some math that takes into account the base 20% chance of no pregnancy by luck, and the fact that other methods have that base rate built in. Perhaps someone will be along soon to elaborate on that math.

In order to take the baseline into account, you’d really need information about how often birth control methods fail regardless of whether those failures resulted in a pregnancy. The data we have already includes the natural rate because it’s only tracking pregnancies.

It is correct that we’re assuming that the birth control methods are more effective than a lucky rabbit’s foot, but that’s likely well-justified by the research literature. If you want a more general approach that can allow for methods that aren’t effective, you might be able to work something out with conditional probabilities and inclusion-exclusion, but it’s likely to be ugly.

It holds as an illustration of the fact that multiplying the “no pregnancy” probabilities overestimates the effectiveness even in the simplest case. Some of the “effectiveness” of the methods in the OP comes from luck too, and it’s a mistake to count that repeatedly. I didn’t deny that three real methods is better than just one; clearly it is.

Sure. Taking your 20% effectiveness of luck, we could say that the sponge decreases the pregnancy risk by a factor of 0.1875 (from 80% to 15%). The factors for the other two methods are 0.125 and 0.0625. Multiplying these factors together gives an overall factor of 0.0014648. Multiplying this by 80% pregnancy with no birth control gives a probability of pregnancy of 0.0011719. This is about 56% higher than what you’d get by multiplying the effectivenesses. Actually, it’s higher by exactly 1/0.8^2, because the simple multiplication counts a factor of 0.8 three times when it should only count it once. If the probability of pregnancy with no birth control were 50%, the difference would be a factor of four. Combining more methods increases the difference between the correct and incorrect calculations.

I think any heterogeneity among couples in fertility or frequency of sex may mess this up, but I haven’t thought it through.

Of course the genuine birth control methods are more effective than luck. I used ineffective methods to illustrate the fallacy of the reasoning being applied for effective methods.

I’m not trying to allow for methods that are not effective. I’m trying to get the right answer for methods that are effective. I believe it can be done as I’ve suggested above, if we make the simplifying assumptions you were making (and doubting) in earlier posts.

I appreciate that some useful maths is going on, with careful examination of what the statistics means etc.

Having said that, what about Angel using a condom with a lucky rabbit’s foot on the end and Buffy taking the pill while holding on to a horseshoe?
Also Angel is a vampire. :eek:

It’s not so much that I’m doubting that the birth control methods fail independently as long as they’re properly used; it’s that I’m not going to make that assumption without some specific reason to think it’s true. Assuming independence just cause it makes the calculations easier is bad form.

Failure rates of contraceptives are calculated using the Pearl Index. It is not risk of pregnancy per user per year of use, it is risk per 100 woman/years.

i.e. If 100 women use the method perfectly for one year, how many will fall pregnant using standard pregnancy testing methods (i.e. positive urinary HCG at 4 weeks after the last period).

If you were to take typical user effectiveness over perfect use, failure rates would increase.

If you were to use a more sensitive pregnancy testing method (e.g. daily serum HCG measurement from day 14 of each cycle), failure rates would increase.

How is this different in practical terms, since if 5 of the 100 women get pregnant in a year, an individual woman is concerned about the probability that she will be one of those 5?