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*Today I'm featuring another guest post from my good friend, Meredith. This short writeup (originally from her blog) demonstrates some basic statistics, and how they might apply to a very real world example. Given the misuse and misunderstanding of these basic stats in the media and current political discussions, and rampant junk science in my Facebook feed, I think this is a timely reminder.... take it away Meredith!*

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**Unlikely things happen all the time.**

Here’s an example. Let’s say you are rolling a 20-sided dice. You probably won’t roll a 20. I mean, you

*might*, but you have a 1-in-20 chance, which is only 5%. This argument works for any number on the dice. Yet, you will roll*some*number between 1 and 20. No matter what you get, it was unlikely… but at the same time, you were bound to get an unlikely result. Weird, huh?
Now let’s say you have a very funny-looking dice with 100 sides on it. Each number only has a 1% chance of coming up. So, let’s raise the stakes a little. Each time you roll, getting 1–99 is just fine. Nothing happens.

*But*, if you roll a 100, you have to pay $10,000.
So, don’t worry! 99% of the time you will be just fine. Just don’t roll the dice any more than you have to—it’s a pretty boring game without any apparent reward, anyway—and try not to worry too hard, because statistics is on your side. Right?

You’re curious, though. You wonder… how many times would you need to roll the dice for it to be more likely to get that 100, just once, than to avoid it completely? If you do the math1, you’ll find that 69 rolls puts you above the 50% mark. In other words, you are more likely than not to get a 100 if you roll 69 times.

Feeling lucky? Want to keep rolling? By the time you’ve rolled that strange 100-sided dice 700 times, you are more than 99.9% likely to get the dreaded 100.

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**Contraception fails much more often than 1% of the time**.

Every time a woman has sex with a man, she rolls a dice. Depending on her contraceptive method of choice, or lack thereof, her dice has a different number of sides on it. But each roll always holds the possibility of pregnancy. Depending on her work, health, and insurance situations, she could be out a lot more than $10,000 in the coming year, not to mention having a child to raise.

**Is your dice a condom?**If you use them perfectly, that’s a 2% failure rate over one year. You only need to roll 35 times to be more likely than not to get pregnant2.

**Is your dice a birth control pill?**If you use them perfectly, that’s a 0.3% failure rate over one year. You need to roll 231 times to be more likely than not to get pregnant2.

This is the absolute best case scenario for these common contraceptive methods. It is why methods like implants and IUDs with extremely low failure rates of 0.05–0.2% are gaining popularity. It is also why emergency contraception exists—think of this as a second “bonus dice” you can roll if you get unlucky with the first one.

We can play this game all day. Women play this game their whole reproductive lives. You can’t take our dice away. You can’t tell us not to roll (well, you can try, but it does absolutely no good). But apparently some employers can deny us access to certain dice and virtually all bonus dice based on a “sincerely-held belief” in junk science.

And yes, women could ignore our employers’ preferences, save our hard-earned money, and go buy whichever dice we like. But

*this*game has a different set of rules. Suddenly we have to be able to afford the dice we want. Suddenly it is not the same game other women can play for free.
Someday, I hope all women (and men!) can have free access to all manner of highly effective, side-effect-free, reversible birth control. I know that doesn’t seem very likely to happen any time soon. But then again, unlikely things happen all the time.

1 The math is actually pretty easy. I’ll use the notation P(something) to indicate the probability that something will happen.

P(not rolling 100) = 99/100 = 0.99

P(not rolling 100, with n rolls) = 0.99n

P(rolling 100, with n rolls) = 1 – P(not rolling 100, with n rolls) = 1 – 0.99n

For this last probability to be more likely than not, it needs to be greater than 50%. So when we solve this equation for n number of rolls:

1 – 0.99n = 0.5

We get n must be 69. In other words, if we roll 69 times, we’re more likely than not to get a 100.

If instead we want to be 99.9% sure of getting a 100, we write it like this:

1 – 0.99n = 0.999

Which tells us n must be 688 (nearly 700). If we roll 688+ times, we are 99.9% likely to roll at least one 100.

P(not rolling 100) = 99/100 = 0.99

P(not rolling 100, with n rolls) = 0.99n

P(rolling 100, with n rolls) = 1 – P(not rolling 100, with n rolls) = 1 – 0.99n

For this last probability to be more likely than not, it needs to be greater than 50%. So when we solve this equation for n number of rolls:

1 – 0.99n = 0.5

We get n must be 69. In other words, if we roll 69 times, we’re more likely than not to get a 100.

If instead we want to be 99.9% sure of getting a 100, we write it like this:

1 – 0.99n = 0.999

Which tells us n must be 688 (nearly 700). If we roll 688+ times, we are 99.9% likely to roll at least one 100.

2 Statistics from this site.

*Note that per-year failure rates are not necessarily the same as per-roll failure rates*. Contraception failure rates are typically calculated as “the difference between the number of pregnancies expected to occur if no method is used and the number expected to take place with that method,” so while this analysis may not be completely sound, the take-home message is unchanged: highly effective birth control is incredibly important.