# Understanding the Long-Term Risks of Contraceptive Failure

An old legend tells of a Western anthropologist researching an indigenous community in the jungle. He was on a journey with some guides when they stopped for the night and he found a nice sleeping spot under a dead tree. The guides were horrified and refused to let him sleep there, citing the risk that a branch, or the whole tree, would fall on him in the night. He thought this was absurd, although he moved to keep the peace: how likely could this possibly be?

Math: If the chance of an event per unit of time is a constant $$p$$, the average time you wait until it happens at least once is $$\frac{1}{p}$$. (This scenario is modeled by the geometric distribution.) So with a 1 in 10,000 chance per night, it takes 10,000 days, or 50 years at 200 days per year, to reach a fifty-fifty chance.

It’s the same story with contraceptives. Contraceptives appear to be pretty effective. People sometimes giggle at those who have unplanned pregnancies, assuming it’s their fault for doing something wrong. And it’s true that user error is responsible for about 80% of failures. But a majority of pregnancies in the US are unplanned, and many of those people were using contraception, so the other 20% is still a lot of people! The problem is that most straight people spend several decades regularly having sex under conditions that could result in pregnancy, and even a small-sounding chance of failure adds up.

Most people don’t have an intuitive understanding of the surprising danger of repeatedly taking small chances; indeed, I think many people don’t even know all of the contraceptive options available to them or the differences in their effectiveness, much less how those differences compound over time. The good news is that, on the flip side, even small differences in contraceptive effectiveness can have massive effects on lifetime risk, so if you understand the risks and choose wisely, you can make things much safer for yourself.

Consider: a 1%-per-year failure rate and a 2%-per-year failure rate both sound low enough for comfort, but you’re twice as likely to get pregnant with a 2%-failure contraceptive. If you’re playing the role of the anthropologist in the legend and only having sex a few times, the difference is in fact pretty much irrelevant because the base rate is small enough to be comfortable anyway. But if you’re playing the locals and you’re married and having sex with someone regularly for years and years, these small differences add up, the base rate rises, and the twofold difference becomes a big deal. (Preview: 1% failure for 10 years adds up to a 9% risk, while 2% for 10 years adds up to an 18% risk; many contraceptive protocols are much worse than 2% failure per year.)

Compounding the problem is that our available data on contraceptive failure is limited; there is no standard model that tries to adjust for differences between individuals, and even the methodology used to test different contraceptives is somewhat suspect at times. I am continually astounded that both public health authorities and otherwise well-informed individuals accept such poor information about their risks and risk factors. An unintended pregnancy may not cause loss of life, but it is often a life-altering event, enormously expensive financially, psychologically, physically, and most of all in opportunity cost. Large-scale plans often have to change dramatically.

Further, recent social trends are increasing the impact of unintended pregnancies. More and more couples are intending never to have children; an unexpected baby is one thing when you already have three kids, or you were planning on your first in another year or two anyway, and quite another when you have zero and never wanted any. People are also marrying later, increasing the size of the window when most people are sexually active but not yet in a position to raise children. Most couples have fewer children, meaning an extra child makes a bigger difference to their plans and the time window during a marriage in which an extra child is highly inconvenient is larger. And abortion laws are becoming more restrictive in many places, removing a possible safety net (albeit one that nobody wants to need). Put together, the Value of Information here is enormous and increasing, but otherwise intelligent people, and their whole societies, are putting an unreasonably low priority on getting all the information they need to make informed decisions.

I assume part of the reason we rarely discuss contraception is that we have to talk about sex, which seems impolite and embarrassing (I guess; this article seems sufficiently clinical that it doesn’t embarrass me at all). I don’t think I’ve ever been part of or witnessed any conversation about contraceptive effectiveness or lifetime chance of failure, in real life or online. But we should get over this. We talk much more about, say, financial planning, career planning, improving interpersonal skills to have better relationships, and even using condoms to avoid STDs. Getting long-term contraception right belongs on this list of boring adult topics that can have a huge impact on your life.

I’m a big believer in the adage that if you can’t find something on the Internet, it’s because you haven’t created it yet, so I went on a research binge for a couple of days to explore the data and the issues involved here. I’ve collected what I learned in this article, along with a simulation spreadsheet, in the hopes of helping others get an intuitive understanding of the math involved here and make intelligent plans about their own future. I hope that someday, some public health agency creates a high-quality model incorporating all this information (and more information that we should be collecting but aren’t) and makes it available to the public, and maybe to some high-school health students.

## Should you trust me?

Let me make one thing clear: I’m a guy on the internet writing on a personal blog. I am not a researcher and I am not paid to study sex or contraceptives.

I am, however, science-literate, have a moderate background in statistics, and care about getting this right. I wouldn’t want you to think this is peer-reviewed research – it is not, although I have tried to base my figures and methodology on such research – but I think it’s probably more accurate and more useful than your received cultural wisdom on contraceptives, whatever that may be, and I’m confident that reading this article will make you smarter about this topic on net. If you’re considering making life decisions on the basis of the information I’m providing here, though, I would, as always, encourage you to do additional research to verify my claims, whether through public health organizations, medical professionals, or more scientific avenues. This is big stuff, and you shouldn’t trust any single person to get it right, least of all me.

I cannot and do not guarantee this article or the simulator is error-free. The statistics involved are better than nothing but always mushy, and either the data I’ve used or my analysis of them could be wrong. And remember that if you’re reading this article because you have the kind of sex that could result in someone getting pregnant, you will never get your chance down to actual zero, so be ready for the possibility that you get unlucky even if I did everything right and you do everything right.

If you do spot any errors or have methodological qualms that you don’t think I’ve adequately addressed, please let me know.

With that out of the way, let’s look at some background information we’ll need to understand exactly what the data and the simulator can and cannot tell us.

## Limitations in data and models

Our models of contraceptive effectiveness are frustratingly limited. Typically, the studies run like this: some couples are assigned to try the new contraceptive (or an old one), they use it exclusively for a year and report how consistently they used it and whether they got pregnant, and then the data is collated and you end up with the percentage of couples who got pregnant while using your contraceptive over the course of a year. Most of the time, people calculate two figures: “typical use,” which is the yearly percentage failure for average people who occasionally screw up the procedure or fail to use the contraceptive at all, and “perfect use,” which is the yearly percentage failure for people who always do everything correctly (you can also think of it as the yearly percentage failure caused by the device, medication, or behavior modification itself failing to prevent pregnancy, excluding any user error).

This format is convenient for many reasons. It’s easy to measure, easy to quote, and easy to understand. But it also flattens everything into an average number, and most couples aren’t exactly average. This means it ends up being a good way to compare methods against each other, but a fairly poor way to measure your personal risk. (Actually, for statistical reasons, it may not be a good way to compare methods either. We’ll talk more about this later.)

Here are a few key things the metric leaves out.

### Change in effectiveness over time

It’s not clear whether the chance of failure is flat if you continue to use some method beyond the first year. It seems reasonable to suppose, for instance, that the chance of messing up a method, e.g., putting on a condom incorrectly, might decrease as you gain experience using it. And if you’ve been sterilized for 10 years without any problems, it seems like you’d have a lower chance of an unintended pregnancy than the first year after the surgery: if no evidence of the surgery being done wrong has emerged and no spontaneous reversal has happened at this point, you’re probably looking at least a little better than before. But I haven’t been able to find any actual data either way. This article says we can assume the chance will remain constant, but doesn’t explain why. Other authors suspect, like me, that we can expect it to decline somewhat, but nobody seems to know by how much.

This article was published in 1988 (seriously, 34 years later we haven’t settled on a better model?) and raises many of the same concerns I am raising here. They ran three models, one in which the one-year failure rate was naïvely extrapolated over 10 years, one in which it fell linearly by half over 10 years, and one in which it fell linearly to zero over 10 years (presumably unrealistic). The differences in final outcome between these models were surprisingly small (although many of the most effective modern contraceptives weren’t available, or weren’t fully refined, at that time; the differences for people on these would be larger in practical terms).

As far as I can tell, almost everyone who runs long-term projections assumes that the chance is flat per unit of time when creating estimates for a single person, and the majority of them extrapolate from the published one-year rate, so that’s what I do too. This means there’s a good chance that my numbers are on the high side, and unfortunately there’s no way to know by how much. Given the choice, though, I’d rather be warned that my contraceptives are more likely to fail than they are than the reverse, and I suspect you agree.

A far more reliable way to do all this would be to run long-term studies of couples using different contraceptives and compile actuarial tables (also called life tables) of the results. While several authors I read recommend such tables as a solution to our statistical woes, as far as I can tell, nobody has created publicly available tables for the periods of time we’re discussing here. I get that this would be difficult and expensive, but it doesn’t seem like the necessary funding would be out of proportion to the importance of this issue. Sigh.

Caution: The measured chance of failure will almost assuredly decrease over time in a longer-term study, when considering the entire group of study participants. However, much of this is caused by selection bias, so it doesn’t necessarily apply from the perspective of an individual. We’ll discuss this in full in “Measurement bias and statistical error,” later.

### Baseline fertility

Some couples are naturally more fertile than others. They have a higher chance of unintended pregnancy – if they have a higher base chance of getting pregnant, and whatever pathway the contraceptive is designed to cut off isn’t cut off, they’re more likely to lose their last saving roll, so to speak.

### Frequency of intercourse

Some couples have more sex than others. It should be unsurprising that people who have sex more often are more likely to get pregnant, contraceptives or not.

I couldn’t find any resources or studies discussing the magnitude of this effect. If you know of any, please point me to them! “Understanding contraceptive failure” (see further reading) points to an older study (abstract) which found that people using a diaphragm and having intercourse more than four times a week were more than twice as likely as people having it fewer than four times to become pregnant in the first year, but the same study found only a 15% increase for contraceptive sponge users. (This data was collected only accidentally during the study in an attempt to control for this factor; the purpose was to compare these two methods of contraception, not the effect of frequency of intercourse on failure. As such, the authors don’t evaluate statistical significance or possible methodological dangers on this point.)

### Age

Owing to the above two factors, the chance of an unintended pregnancy ordinarily decreases as people age.

### Cyclical change in fertility (?)

Women are not even close to equally fertile throughout their menstrual cycles. This means that some days are dangerous and others are almost completely safe, if some kind of failure occurs (indeed, paying attention to this cycle is a moderately effective contraception method).

Perhaps this doesn’t matter because the times people have sex are randomly distributed across the menstrual cycle, or if they’re not the distribution is similar between couples. But given that many women are more interested in sex at their most fertile times, I wonder if some couples might not experience this effect more strongly than others (and go along with it more often) and thus be at measurably greater risk despite an identical average frequency of intercourse.

To be clear, I am making this one up – unlike for the other factors mentioned here, I couldn’t find any researchers who have made this hypothesis. So I may be wildly off-base here; then again, this would be challenging to measure even if it exists. I mostly wanted to include this pet hypothesis to emphasize how complicated the real picture is compared to the simplicity of our data. As Nassim Nicholas Taleb is fond of pointing out, life is far more complicated than a game with fixed probabilities; we can throw out measured probabilities until the cows come home, but in real life there will always be other factors we don’t know about and can’t account for, so we just have to do the best we can (and be prepared for the worst possible outcome – if we can identify it).

### Measurement bias and statistical error

As with any field of study, the design of contraception studies can be poor and the results can be misinterpreted. It’s not clear that all available data is of high quality.

Here’s one particularly insidious bias. While it is generally assumed that the actual risk of pregnancy for a given couple with constant fertility and frequency of intercourse remains the same over time, the total pregnancy rate per unit of time for a study group will usually decline over the course of a longer study. This happens not because each participant’s risk declines (at least, not necessarily), but because these studies only measure up to people’s first unintended pregnancy. People who are less careful with the method or were more likely to get unlucky due to fertility level and/or frequency of intercourse are more likely to drop out of the study early, causing the reduced sample going forward to have a lower overall chance of further pregnancies. This doesn’t mean the figure is wrong as a description of the group, but if a study runs for 10 years and provides its ten-year figures, those ten-year figures don’t necessarily apply to you as an individual using that method of contraception for ten years. (More on this in the “Flat rate across time” assumption section, later.)

A corollary is that studies run over different periods of time and presenting a “percent pregnant per year” metric are not comparable. Running a study over a longer period of time will cause the measured average chance of pregnancy to decrease, for the reasons described above, even when the actual effectiveness for any individual is unaffected by the length of the study. (More useful information in “Probability of failure over time” and “Factors that influence efficacy,” here.) Again, we could improve on this by using actuarial tables rather than a percent-per-year metric.

Another bias can occur when a study aggressively pregnancy-tests participants. Since a substantial number of pregnancies are lost before the woman knows she’s pregnant, this can make the effectiveness appear much lower than in typical studies that don’t apply regular testing (and so don’t count early miscarriages as pregnancies). Of course, neither approach is strictly wrong; counting pregnancies differently just makes the effectiveness numbers incomparable.

### Conclusion

Not only do we not have a standard model or rule of thumb that takes into account factors like these, I haven’t seen anyone try to gather the necessary data, for instance with subgroup analyses on their studies. Admittedly, I haven’t looked all that hard – probably there’s some obscure article somewhere that I didn’t find by searching – but this seems like a pretty obvious thing that we should be trying, and in a world that matched my expectations, such results would be trivially available to the public in an easy-to-understand form. I’m particularly stumped by the lack of any way to adjust for how much sex a couple is having, given how drastically people differ on this scale. Even breaking down by age groups would seem to be a super-easy, if imperfect, proxy for fertility and frequency of sex, and not all that difficult to do.

But for now, we have the data we have, and it’s certainly a whole lot better than nothing.

## Assumptions

Let’s talk about a few more assumptions I’ve had to make in my simulator, and how they might affect your interpretation of the results.

### Flat rate across time

We discussed this in “Change in effectiveness over time” – the simulator assumes that the effectiveness of a contraceptive is equivalent for a particular couple at every unit of time, regardless of how long they’ve been using it. This seems to be widely accepted practice and, while it’s not necessarily correct, that’s what we’re going with here.

If there were a bias created by a flat-rate-across-time treatment, it would likely be in the direction of making failure probabilities appear higher in the simulator than their actual values.

### Average couples

As described in “Limitations in data and models,” above, all available effectiveness statistics are averages among all couples who participated in a study. You should thus interpret risk figures this way, too: not your exact chance, but the chance of an average couple who used this protocol. That means that if you want to know your personal risk, and you think your baseline risk is lower or higher than an average couple’s, you need to adjust the figure accordingly. See “The multiplier,” next, on suggestions for doing this.

For the same reason, the simulator makes silly assumptions surrounding infertility. It caps the per-year risk of unintended pregnancy at a flat 85%, generally accepted as the chance of getting pregnant when using no contraception at all. But in reality, if you have unprotected intercourse for 5 years and never get pregnant, you definitely wouldn’t expect your chance in the sixth year to still be 85%. If you feed this scenario into the simulator, it will tell you your chance of an unintended pregnancy after 5 years is 99.99%, which is obviously untrue (for starters, far more than one in ten thousand sexually active people are totally infertile). For this reason, you should expect extremely high Interval Risks (risks accumulated over multiple years) or Lifetime Risks to be somewhat inflated. Again, better this way than the reverse.

Note: You may also see a figure of 40% per year for likelihood of pregnancy with unprotected intercourse floating around. It seems that 85% has been measured in contexts where people are trying to get pregnant, or at least not trying not to get pregnant (e.g., newly married couples in societies where contraception is not commonly used), while 40% has been seen in whole-population contexts, where some people are already pregnant, some people rarely have sex, and so on. The true figure we want here may be somewhere in between, but I have no way to figure out how to blend these figures accurately, so I’ve again chosen the more dangerous one.

### The multiplier

In an attempt to help you explore how factors like overall fertility and frequency of intercourse might affect risks, I’ve added a column called Multiplier to the simulator, which allows you to increase or decrease the yearly risk by a fixed factor on top of the base failure rate suggested by your selected contraceptive methods. For instance, if someone is at peak fertility in their late twenties and having sex every day, you might guess that this might make them half again as likely as an average sexually active person to get pregnant, and so fill in 1.5. Since you and I are both guessing at all multipliers we try, don’t expect this column to allow you to produce a perfectly accurate number for your own sex life, but it may be useful to compare how much the figures change depending on different reasonable guesses of how someone differs from average.

### Independence

The simulator assumes that methods of contraception used concurrently are independent – that is, the chance of one failing is entirely unrelated to the chance of the second failing, and vice versa, so that risk reductions perfectly “stack” on top of one another. (See the Swiss-cheese model of risk: if the first layer fails, the next layer takes over, and the holes have to line up for there to be a failure.) For the most part this seems like a reasonable assumption, but it could be wrong in at least two straightforward ways.

1. You could pick methods that don’t make any sense together. For instance, one woman can’t use both a hormonal implant and a hormonal IUD to decrease her risk of an unintended pregnancy to 0.001%/year, and one man can’t simultaneously use a condom both “typically” and “perfectly” to further decrease his risk, but the simulator will let you choose these combinations and cheerfully multiply out the resulting risk reduction. To get meaningful results when selecting combinations of contraceptives, you need to understand what the selected methods are and how they work, and use common sense.

2. If dealing with “typical use” figures, there may be environmental or social factors that mean both methods will tend to go unused or be used incorrectly at the same time. For example, if a couple normally uses both spermicide and a condom, and they decide to have sex unexpectedly while away from home, they are presumably more likely than chance would suggest to be lacking access to both methods of contraception.

The simulator also doesn’t support combinations of methods used separately, rather than together (e.g., you have three available methods and commit to always using at least one at any given act of intercourse). I think there are too many factors here to be confident that any result obtained by mathematical manipulation, rather than a study of this specific behavior, is even mostly correct.

## Selected conclusions

To get you started, here are a few simple conclusions from the available data and my playing around with the simulator that might surprise you. Remember that these conclusions are subject to all the provisos in the sections above!

• Condoms are less effective over the long term than you probably think. Average couple, 30 fertile years, using a condom perfectly every time: 45% chance of at least one unintended pregnancy. If condoms are only used “typically,” the simulator says 98%. (Realistically this will max out somewhat lower, as noted in “Average couples”, but regardless, that’s a scary number.)

• That figure drops dramatically with better contraceptives. A perfectly used garden-variety combined birth-control pill reduces the 30-year risk to 8.6% (keep in mind that perfectly using the pill is hard), a vasectomy to 3.0%, and the implant to 1.5%.

• Combining independent methods, even sloppily used ones, can be a highly effective strategy. The pill and condoms, both in typical use, together reach a respectable 30-year rate of 24%. This isn’t a fantastic figure, but the combination transforms two protocols that are both stupidly unreliable over a 30-year period (either alone reaches upwards of 85% failure) into a nearly-acceptable one. Swapping one of the methods for a more effective one yields a very strong protocol; a copper IUD used perfectly by an average couple for 30 years has a 16.5% total risk of failure, but adding condoms in typical use (allowing for occasional unprotected sex if you forgot them or aren’t feeling it) is enough to drop the 30-year risk to 2.3%. For couples with low risk tolerance and high confidence in their life plans, a double sterilization (vasectomy + tubal ligation) reduces the 30-year risk to 0.015%, or about 1 in 6,700. To put that into perspective, that’s a bit more than twice your lifetime risk of being struck by lightning (1 in 15,000, according to NOAA).

Just to emphasize how underappreciated these differences are: before reading this article, would you have been able to say that some birth-control protocols have a 98% chance of unintended pregnancy over a lifetime, while others reduce it to 0.015%? I’m sure you would have correctly answered that having both partners sterilized is more effective than inconsistent condom use, but would you have guessed it was a 650-fold difference?

• It’s true that withdrawal isn’t very effective in typical use, but condoms in typical use aren’t all that much better. Average couple, three years of typical-use withdrawal: 48.8%. Three years of typical-use condoms: 34.2%. (Just three months of typical-use condoms: still 3.4%. That’s more than the 30-year risk of some properly used, highly effective options!) The most important contraceptive-related lesson we should be teaching in sex-ed is not that withdrawal doesn’t work and condoms do, but that if you’re having more than a brief fling with someone, you’d best follow the directions or you’re probably going to regret it. (And even three years of perfectly used condoms is a 5.9% risk. If you wouldn’t be pleased to end up pregnant, try to find something better.)

Fermi Estimate: If you’re a sexually active straight guy who hasn’t settled on a partner yet and you aren’t sure you’re using condoms more or less perfectly, reading the instructions on your condom package and then following the correct protocol from now on might be the best financial decision you’ll ever make in your life. You can save literally thousands of dollars an hour – tax-free – in average child support, not to mention avoid all sorts of other problems, by going from typical use to perfect use. Please do this.

Full estimate: On average, it’s somewhere around $200,000 to raise a child in America (low end). You pay half:$100,000. Perfect condom use cuts the risk of an unintended pregnancy by 5 times over typical condom use (low end). Baseline no-instructions lifetime risk of fathering a child out of wedlock unintentionally: 20% (high end? depends a lot on your behavior). New lifetime risk after reading instructions and committing to follow them: 4%. 16% chance of saving $100,000 over 15 minutes =$64,000 per hour.

• And actually, perfectly used withdrawal isn’t all that bad according to most estimates. I’m not suggesting you should rely on this as your only method of birth control; it’s not all that effective at baseline, and it’s probably difficult to do perfectly every time. But even done typically, it’s more than four times as good as nothing (typically used condoms are about seven times as good). Should you find yourself in a situation where you have no other options and are not feeling level-headed enough to call off your tryst, you should definitely use it! And it could be profitably combined with another method.

Note: Because I can already hear people yelling at the least charitable interpretation of my comments above, I am not anti-condom by any means. Condoms are great: they’re trivially easy to use even on a moment’s notice, cut your risk by a lot, and also protect against STDs, which is a big deal if you’re in a lot of short-term relationships. But if you’re straight and in a long-term relationship, please don’t assume they work miracles against pregnancy long-term – which you might if you never look into it. They don’t (in fact, I’d be inclined to say that stating they “work” at all over a 30-year period is misleading, if your goal is to avoid any unintended pregnancies), and you should strongly consider additional measures if you’d be unhappy to end up pregnant.

## Using the simulator

In reality, your lifetime plan will probably be more complicated than the samples presented above. You may have more than one partner, you’ll get a brief respite from the statistics if you’re young and want children at some point, and you probably won’t find the same contraceptives convenient for your whole life. Plus, by the time 30 years have gone by, we’ll probably have new options to choose from. (Effective reversible options for men? Pretty please?)

For this reason, I’ve created a spreadsheet to allow you to simulate any plan you like and easily make hypothetical changes and see the impact. You can include up to ten time intervals of any length, each using any combination of contraceptives and multiplier, and see the total lifetime risk all at once. (You can also use different intervals to contrast different options, ignoring the total risk.)

The simulator is published as a read-only Google Sheet. You’ll need to log in to Google Drive and make a copy of it; instructions on doing this are in with the rest of the use instructions on the Instructions tab.

To tie everything together, here’s a video demo in which I use the simulator to assess the lifetime risks of a hypothetical man. (Sorry, apparently I can’t make short videos about things I’m interested in. I didn’t expect this to take 25 minutes.)

## Simulation methodology

You might wonder how exactly the simulator calculates its probabilities. Here’s a detailed rundown. This section is optional – if you aren’t into math or stats, you don’t need to read this section to understand everything.

### Data

My figures were obtained from Wikipedia’s Comparison of birth control methods article; I spot-checked some of the figures in the original sources. While there may be a greater chance of Wikipedia having a set of studies whose numbers aren’t directly comparable than if I had picked a particular author’s analysis, and it’s always possible someone wrote the wrong number there, I felt that as a layperson I had a greater risk of selecting a uniformly worse source, or an out-of-date series of figures, if I tried to pick a set of studies or even a summary directly from some paper. If you don’t like one or more of my data points and want to use someone else’s, simply change the relevant figures on the Effectiveness tab of the spreadsheet and it will automatically propagate into the simulator.

As I noted earlier, if you use the simulator to select contraceptive methods, you should go do more detailed research on whatever looks good in the simulator before making a final decision; not everyone agrees on all of the numbers, and sometimes different brands or varieties of a given method have meaningfully different effectiveness statistics.

I eliminated some methods from this list, for the reasons noted:

• Forschungsgruppe NFP symptothermal method: This is a specific type of time and symptoms-based behavioral method. It seemed overly specific and the first cited source didn’t contain the relevant information or (as far as I could tell) any reference to the method at all, which made me suspicious of the integrity of this entry.
• All discontinued and unapproved methods listed (Essure, cervical caps and spermicide, testosterone injection).
• Ormeloxifene: Currently only available in India, so I thought this would be more confusing than helpful to most readers.
• Emergency contraception: This isn’t a general-purpose contraceptive (but of course you should use it if appropriate!).
• Generalized “calendar-based methods”: I was concerned about the quality of the data, and this covers a wide variety of methods with different effectivenesses.
• Abstinence pledge: This isn’t a method of contraception in a sense useful to us here.

The remaining effectiveness rates are tabulated in the Effectiveness tab of the spreadsheet, and they become columns in the simulator.

### Calculating the risk per year

To calculate the risk of pregnancy per year, the per-year failure rates for all selected contraceptives are first multiplied together (if no contraceptives are selected, the value is 1). This works because we are assuming they are independent. To calculate the probability of two independent events both occurring (the first method failing AND the next method failing), the probabilities are multiplied together.

This combined risk number is then multiplied by the Multiplier, if present.

Finally, if the result is higher than 85%, it is capped at 85%, under the assumption that contraceptives never fail so badly they make someone more likely to become pregnant than unprotected intercourse. (It’s possible to reach this level by selecting no contraceptive methods or by adding an unreasonably high multiplier.) This isn’t a perfect solution; 85% is the average figure, so some couples presumably have a >85% chance of pregnancy from a year of unprotected intercourse. A better way would be to allow the number to rise over 85%, but never more than 100%, somehow mapping all reasonable figures higher than 85% onto a range between 85% and 100%. But since 85% in one year is already unconscionably high if you’re trying to avoid pregnancy, and any multiplier high enough to cause the chance to hit the cap if you’ve selected any contraceptive method at all is probably unreasonable, I figure this isn’t worthwhile.

### Calculating the interval risk

First we determine the number of years in the interval. If a raw number was given, that is used; otherwise, if a start age and end age were used, the start age is subtracted from the end age to yield the total number of years (the end age is exclusive of that year of life). We then calculate the probability of failure for the entire interval using the formula

$F = 1 - (1 - p)^Y$

…where $$F$$ is the combined probability of failure, $$p$$ is the probability of failure for one year, and $$Y$$ is the number of years. If you know enough about probability that this immediately makes sense, feel free to skip to the next section – otherwise you can check the derivation below.

We treat the total risk over an interval of $$Y$$ years as a series of independent failure events, one for each year. To combine these failure events, we’ll temporarily consider instead the probability of success, which will allow us to multiply them together to get the total chance of no failures (since we assume the chance of a failure each year is independent of the others). From there, we can easily calculate the total chance of not having no failures, i.e., having at least one failure.

The complement of an event $$E$$, written $$E^C$$, is the event that $$E$$ does not occur; its probability is 1 minus the probability of $$E$$:

$P(E^C) = 1 - P(E)$

So put more mathematically, to calculate the total failure chance $$F$$ over $$Y$$ years, we calculate the probability of the complement of the product of the probabilities of the complements of the failure events $$E_i$$ for each year $$i$$ – i.e., the chance that it does not happen that in every year contraception succeeds in preventing pregnancy.

\begin{align} F &= 1 - \prod^{Y}_{i=1}\: P(E^C_i) \\ F &= 1 - \prod^{Y}_{i = 1}\: (1 - P(E_i)) \\ F &= 1 - \left[ (1 - P(E_1)) (1 - P(E_2)) \cdots (1 - P(E_Y)) \right] \end{align}

Since in our case the events $$E_i$$ for all years $$i$$ have the same probability, this simplifies to:

$F = 1 - (1 - P(E_1))^Y$

…Q.E.D.

The lifetime risk treats each simulated interval as an independent event and multiplies their complements together using the same basic formula, only this time we can’t take the shortcut of inserting one probability and taking it to the power of the number of years, since the probability for each interval can differ. That leaves us with:

$F_{\text{total}} = 1 - \prod^{|\text{Intervals}|}_{i = 1}\: (1 - F_{\text{interval}\:i})$

…which we implement directly in the spreadsheet.