Contraception Failure

There is a New York Times article showing cumulative contraceptive failure rates over time, i.e., if you use a certain contraceptive method for five years, how likely is it that you would have become pregnant in that time? Here is how the values were calculated: “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.” By “success rate,” they mean percentage of women who avoid pregnancy in one year. However, it is important to remember that these rates are in fact only for the first year of use. The question becomes whether these rates can reasonably be applied to later years.

I have found a possible answer in a 2004 article by James Trussell, ‘Contraceptive failure in the United States’:

We confine attention to the first-year probabilities of pregnancy solely because probabilities for longer durations are generally not available. There are three main points to remember about the effectiveness of contraceptive methods over time. 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). 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 or those who are relatively infertile or who have lower coital frequency. This decline will be far less pronounced among users of those methods with little or no scope for imperfect use. 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 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.

As the final sentence indicates, what the New York Times authors have done is mathematically correct. However, they should not have been so quick to take the first-year “success rate” and assume it is the same in the following years. The first-year rate includes women who, for one reason or another, were unsuited to use that kind of contraception (e.g., due to difficulty in regimen adherence, physiological reasons, etc.) and became pregnant and stopped using it. It follows that the second-year “success rate” will be greater, since it no longer includes these users. However, one might predict the reverse: perhaps the second-year “success rate” will actually be lower, since users who did not get pregnant in the first year might become more complacent about adherence.

Quickly searching, I was able to locate one study that measured non-injection hormonal contraceptive failure rate over three years (this category includes not just oral contraceptives, but also patches and rings). The cumulative failure rate was 4.8%, 7.8%, and 9.4%, after one, two, and three years, respectively. As one can see, the failure rates after two and three years were lower than what would be predicted based solely off the first year number (predicted: year 2, 9.3%; year 3, 13.7%). Although a look at the numbers for injectable (DMPA) contraception might give us caution: 0.1% after year 1, 0.7% after year 2, 0.7% after year 3. The number after the second year is in fact higher than the predicted value based on the failure rate after one year (predicted: 0.2%), but then after that, there were no additional contraceptive failures. I am not sure what the answer for this is, but it should give us pause about relying too much on this study. While I can admit that the study suggests that it’s common for contraceptive success rates to rise after one year of use, I cannot say this is certain. Finally, the cumulative failure rates for IUDs and implants over three years were: 0.3%, 0.6%, 0.9%. This matches up with the predicated values based off the first year (predicted: after year 2, 0.5991%; after year 3, 0.897%), probably because user adherence is hardly an issue here.

Cumulative contraception failure is an important area of research that seems neglected, given that “[t]he typical woman who uses reversible methods of contraception continuously from age 15 to age 45 would experience 1.8 contraceptive failures. If we consider both reversible methods and sterilization, the typical woman would experience only 1.3 contraceptive failures from age 15 to 45” (from the Trussell article). While it may be mentioned that there are much more effective methods (e.g., IUDs or implants) than what is commonly used, I find it very unfortunate that the one method with a perfect success rate often gets ignored or dismissed as impractical, especially given its role in developing certain virtues.



At the long term care facility at which I used to volunteer, bingo is played every other week or so. In the room in which it is held, there are six tables (numbered 1-6 in the sketch below), each holding about four players. There are also some individual seats for extra players. Each player receives two bingo cards, and bingo is usually played according to normal rules, with the exception that getting four corners on one of the bingo cards counts as a bingo as well.


Among other things, I was often responsible for distributing bingo cards to the players. I noticed that the two bingo cards I gave to a player typically carried a very similar series of numbers. I began to wonder if I could increase a player’s chances of winning if I gave them two cards with as minimal overlap as possible (and perhaps by doing so, decrease the number of balls the caller has to draw in order to reach bingo). I figured wins would occur more quickly with a diverse pair of cards, for the same reason that you gain no advantage in the lotto when having two identical sets of numbers.

Running the numbers
I made a simulation using Java, which I checked by comparing the numbers it generated with the probabilities calculated on this site. After verifying, I set up a game with thirty players, each having two randomly generated cards. I then ran 10,000 games to determine the most likely ways to reach bingo. The 10,000 games resulted in 12,051 bingos (more than one bingo can occur per game if two or more players reach bingo at the same time). The data is as follows:
running the numbers

Note the following:

  • 59% of bingos used the free space. This is especially notable since there are more ways to get a bingo without a free space (9) than there are with a free space (4). If we exclude from the analysis the numbers from four corners bingo, the percentage of bingos that did not use the free space falls from 41% to 32.5%.
  • Approximately 25% of games resulted in multiple bingos.

Is a lack of card overlap advantageous?
Next I made sure one of the thirty players had no overlap whatsoever between his cards (call him player X). All the other members had randomly generated cards, which almost always contained some overlap, usually 4 or 5 numbers overlapping between cards on average. I ran one million games. The percentage of winning bingos belonging to player X was, on average, 3.37%. To be more specific about the method, I ran 100,000 games at a time, with player X having the same pair of cards for all 100,000 games, and all other players getting randomly generated cards each match. I ran this ten times for a total of 1,000,000 games. The upshot of all this is that there is no practical benefit for a player to have no overlap between his cards, as any given player has a 1/30 = 3.33% chance of winning a game of chance involving 30 players.

Does card overlap increase time needed to reach bingo?
The above results suggest that card overlap has little effect on time to reach bingo. To confirm, I ran 100,000 games where all players had a purely randomly generated set of cards and another 100,000 where each player had zero overlap between his cards (though this doesn’t rule out the inevitable overlap among different players’ cards). The results showed no difference in number of balls drawn to reach bingo (approximately 18.0 turns in either case).

The answer to all my initial questions was “no.” Practically speaking, overlap on the cards has no effect on the outcome of the game. One reason for this is that, unlike the example of the lotto numbers, the location of the number matters. Having the same number on both cards is not always a mere repetition because the ‘value’ of a bingo number depends on the numbers around it and, also, an overlapped number may fall on a different spot in the two cards.

I think the simulation is solid, given that I verified it independently. The only possible limitation I can think of is that, in the real life situation, there are only a limited set of cards, whereas the simulation randomly generates each pair – but I do not see how that would affect my results. Another possible issue is that player X’s cards were not randomized every game, but, again, I do not see how this would be a big issue, since the balls were randomly drawn each game and all the other players had cards randomly generated each game. It does raise an interesting question though: when we want a random simulation, should we make all parts random, or is it enough to have just one part random? I can share the source code for the simulation with anyone who wants it.