Writing for the New York Times, Gregor Aisch and Bill Marsh model long term failure rates of birth control methods by extrapolating the one-year failure rate based on a simple mathematical formula:

The probability that a woman doesn't get pregnant at all over a given period of time is equal to the success rate of her contraceptive method, raised to the power of the number of years she uses that method.

Similarly, Dr Stephen Genuis models the long-term failure rate of condoms in a Catholic Education Resource Center article

The effectiveness of a contraceptive method is defined in terms of the phrase "number per 100 woman-years." This definition is designed to complete the sentence:

"Of 100 typical users who start out the year employing a given method of contraception, the number who will be pregnant by the end of that year will be _________."

After reviewing the extensive literature on contraception, some variation in results is found. Reported failure rates for condom use vary from about 2 to 35 unplanned pregnancies per year, but a conservative consensus reveals a rate in the range of 8 failures per 100 users each year in the general population. Simple mathematics would conclude that after five years, the number pregnant with this method would be five times the yearly rate. Thus, after five years of condom use, there would be about 40 pregnancies in this group of 100 real people; after 10 years there would be 80 pregnancies.

Thus he predicts a 40% failure rate over five years of condom use.

The mathematics exercise book, One Thousand Exercises in Probability, makes a similar prediction in a probability exercise based on unsubstantiated estimates of a 10% annual failure rate, leading to 41% failure rate over five years.

Wikipedia states that the typical use first-year failure rate is 15%.

However, these simple mathematical models and may not represent reality. For example, significant correlation in failure rates for individual subjects across years, or significantly time-varying failure rates, would invalidate this simple model.

What is the typical use five-year failure rate for condoms?

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    I'm not sure what you're asking for here. We've measured the failure rate experimentally, it's 85% for typical use and 98% for perfect use.
    – Publius
    Mar 28 '14 at 12:15
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    @Oddthinking The claim is that the seven-year (and by extrapolation, using the same reasoning, five-year) condom failure rate is > 1/2. The claim is published in a book. Is that not enough for notability? Similar claims can be found online. Mar 28 '14 at 14:38
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    @Konrad: It is the nature of the book: maths exercises tend to be based on over-simplistic or often fanciful examples. They aren't serious claims, but invented scenarios to practice on. I was hoping similar claims could be found, and your example works for me.
    – Oddthinking
    Mar 28 '14 at 15:13
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    @Oddthinking I accept your argument but I’d like to point out that statistics text books (at least the ones I know) are a bit different in that regard: since an important goal of these books is sharpening people’s ability to assess uncertainty in real life, they tend to choose examples carefully and realistically (many of them are real case studies). Mar 28 '14 at 15:59
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    You can find a visualisation of the failure rates for several contraceptives here. The source states that 63 of 100 women become pregnant within 5 years, when the male uses a condom (typical use). With perfect useage of the condom, the pregnancy rate is reduced to 10 Dec 5 '14 at 13:14

To properly answer this question definitively, we would need the results of a five-year long study. That would be very expensive and of limited value, and I was doubtful that one would exist. Proving the non-existence of studies is practically impossible, but a technique that we have accepted here before at Skeptics.SE is finding an appropriate expert who has performed a literature search and who explained that such studies are missing or lacking.

Fortunately, the 2009 book Contraceptive Technology by James Trussell and Anita L. Nelson, M.D., 19th Edition, covers this topic.

In Chapter 3 (page 28) they explain:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available.

So, as of 2009, there is no definitive answer to your question...

... however, as a bonus, the authors do go on (in the same passage) to warn that the simplistic mathematical models being proposed in the question are inappropriate. [Emphasis theirs, paragraph breaks mine.]

There are three main points to remember about the effectiveness of contraceptive methods over time.

The first and last points support the simple multiplication method, with caveats.

First, the risk of pregnancy during either perfect or typical use of a method should remain constant over time for an individual woman with a specific partner, providing that her underlying fecundity and frequency of intercourse do not change (although it is possible that the risk for a woman could decline during typical use of certain methods because she learns to use her method correctly and consistently).

The second point, however, shows it to be overly-simplistic.

Second, in contrast, the risk of pregnancy during typical use of a method will decline over time for a group of users, primarily because those users prone to fail do so early, leaving a pool of more diligent contraceptive users, those who are relatively infertile, or those who have lower coital frequency. This decline will be far less pronounced among users of those methods with little scope for imperfect use. The risk of pregnancy during perfect use for a group of users should decline as well, but this decline will not be as pronounced as that during typical use, because only the relatively more fecund and those with higher coital frequency are selected out early. For these reasons, the probability of becoming pregnant for a group of users during the first year of use of a contraceptive method will be higher than the probability of becoming pregnant during the second year of use.

Third, probabilities of pregnancy cumulate over time. Suppose that 15%, 12% and 8% of women using a method experience contraceptive failure during years 1, 2, and 3, respectively. The probability of not becoming pregnant within 3 years is calculated by multiplying the probabilities of not becoming pregnant for each of the 3 years: 0.85 times 0.88 times 0.92, which equals 0.69. Thus, the percentage of becoming pregnant within 3 years is 31% (=100% - 69%)

The lesson here is that differences among probabilities of pregnancy for various methods will increase over time. For example, suppose that each year the typical proportion of women becoming pregnant while taking the pill is 8% and while using the diaphragm is 16%. Within 5 years, 34% of pill users and 58% of diaphragm users will become pregnant.

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