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.
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?