Failure rates are determined during clinical trials in smaller populations where they also survey compliance to use of the contraceptives and factors that affect compliance. So these rates are obviously not completely accurate in all populations/contexts.

The failure rate is the difference between the expected number of pregnancies (per 100 people) with no contraceptives and the expected number of pregancies (per 100 people) correctly using that contraceptive. The efficacy rate is 100 minus the failure rate.

So a failure rate of 2% means an efficacy rate of 98%.

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Also note: A 2% or 4% chance of pregnancy is pretty good odds using only condoms for 1 year. If using only condoms for 10 years that number is creeping up on 20% or 40%.

This is a weird one as well because you aren’t taking into account fertility. Say someone gets into a serious relationship at 25, if your’re a women your fertility starts to decrease slightly in your late 20s and even more so after you hit 35.

I don't think the 10 year part is right. After one year 96% of people haven't had a failure. By year two 96% of those remaining so it goes 0.96^10 = 0.66 or 34% failure rate in 10 years in your worst case and 0.98^10 = 0.82 or 18% for your best case.

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Also, keep in mind that the publicized control failure rates represent failure over a 1-year period, but the longer you go out, the higher the failure rate. For example, for male condoms, the failure rate of perfect use is 2% in 1 year, but 18% over a 10-year period. For typical use, it’s 18% after 1 year and a whopping 86% after 10 years. This article has interactive charts that show how the failure rates with perfect and typical use are compounded over time for several different types of birth control:

https://www.nytimes.com/interactive/2014/09/14/sunday-review/unplanned-pregnancies.html

I know that’s not exactly what you asked but I like to throw out this information because I think it’s important.

That's just pretty basic math though:

1-(1-0.2)^10 = 0.18

1-(1-0.18)^10 = 0.86

etc.

I’m an immunologist; we don’t do math.

Kidding, kidding, I understand, but I don’t think it’s intuitive to most people that it works like that. A lot of people think that the quoted failure rates (ie, 2% for male condoms) are the failure rates over a lifetime of use, and the graphics in the article help hammer the message home.