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# Just how dangerous is that motorcycle? The surprising math of recurring risk

There's a huge difference between riding on the back of a motorcycle once, and buying a motorcycle to ride every weekend, or between buying a cookie, and buying a box of 25 cookies, or between taking a boxing lesson once, and sparring regularly.

The difference, of course, is in scope. Even if you have a clear sense of the benefits and risks of doing something one time, the sum likelihood of experiencing these benefits and risks changes as you repeat the activity over and over. Unfortunately, our minds don't necessarily appreciate this important difference between doing something one time and doing something many times. This week we'll be discussing how repeating a risky activity often leads people to misjudge just how much risk they're exposed to.

Typically, the riskiness of the decision actually scales up in proportion to our prediction for T, the number of times we'll perform the activity in question. The problem is that how "significant" the decision feels to us may NOT scale up proportionally to T. This leads people to miscalculate the "weightiness" or seriousness of risky decisions. Consequently, people tend to worry too much about doing a risky thing once, but fail to worry nearly enough about doing the same thing T times.

Consider the basic mathematics of repeated risks. When we do some activity that has pretty low risk "R" each time we do it — say, a 1-in-1000 chance of serious injury — and the risk each time is independent from the previous times, then if we do this activity once, we have a risk of R. By contrast, if we do the activity twice, we face the risk of the bad outcome twice, so the risk is about 2*R. Likewise, the total risk we take on when we do the same activity T times (assuming T is not super large) is about T * R. This is the basic risk formula you should keep in your mind.*

So if you're about to commit yourself to do something 30 times, you're taking on about 30 times the risk you'd take on by doing it once. Does your brain properly appreciate that huge multiplier? Do you feel on a gut level that doing something 30 times more often is 30 times riskier? Probably not. As the risk R gets bigger, individual decisions of course becomes more significant. But the actual risk is about T * R, so R itself is not enough to consider if you plan to repeat the activity a number of times. If we limit our considerations to R by itself, then we're likely to end up too risk-averse on individual decisions, and not nearly cautious enough about repeated choices.

Let's consider motorcycles. According to the National Highway Traffic Safety Administration, American motorcyclists in 2006 faced about a 1-in-26,000 chance of dying for every 100 miles' travel. For reference, this is 11 times the chance of dying that an American 30-year-old man faces on a random day, or 24 times the chances of death that a 30-year-old woman faces on a random day.

So, if you are considering buying a motorcycle, and predict you will ride it for 100 miles each week for the next 5 years, that's about 52 weeks * 5 years = 260 times that you'll go on a 100-mile ride. This behavior would amplify your risk of death by 260:

R * T = (1 / 26000) * 260 = 1 / 100

That is, you've just taken on a predicted 1% risk of death by buying and regularly riding this motorcycle! Of course, injuries are much more common than death (about 20x higher, it seems) in motorcycle accidents, so the chance of injury you've taken on is even higher than that. So this is a very weighty decision. It becomes weightier still if you think you'll ride it for 10 years instead of 5. But even in the 5-year case, consider: how much money would you pay NOT to have to roll a 100-sided die where death occurs if you roll a 100, and a potentially grave motorcycle-accident injury occurs if you roll, say, an 80 or higher? That's the kind of risk you're taking on by choosing to ride a motorcycle 100 miles a week for 5 years.**

Part of the reason we can underestimate risk is because we convince ourselves that we can stop at any time. For instance, we can buy the motorcycle and then later just decide to stop using it. Surely we can, but the question is, will we? We need to make the decision based on what we predict will happen, not what could theoretically happen.

The point is here that doing a "risky" thing once is far less risky than doing a risky thing many times — about T times more risky! — but your brain won't necessarily perceive the right relationship. For small independent risks, the total risk of a bad thing happening scales with the number of times T you do the thing. So remember the risk formula: R * T, and teach it to your brain.

Another clear illustration of this idea can be found in this New York Times analysis of the effectiveness of various forms of birth control over time. Consider the oral contraceptive pill, for instance; the Times analysis finds that 9 women in 100 would develop an unplanned pregnancy after a year of typical use of this popular birth control method (Evra patches and NuvaRings are other hormone-based birth control systems with failure rate similar to the pill):

After 5 years of typical use, the proportion of women who would likely experience an unwanted pregnancy rises to 38 in 100

And after 10 years, the proportion of women on the pill who could expect an unwanted pregnancy rises to a somewhat astonishing 61 out of 100. (The male condom, a similarly popular birth control device, rises to a somewhat astonishing 86 in 100 unwanted pregnancies after 10 years of typical use):

Fortunately, long-acting reversible contraceptives (LARCs) are also available as much more reliable methods of birth control. In 10 years of normal use, the rates of unwanted pregnancy using these methods range from 1 out of 100 (hormonal implants), 2 out of 100 (hormonal IUDs), and 8 out of 100 (copper IUDs). However, according to analysis performed by the Guttmacher Institute only 11% of women who use birth control choose one of these more reliable and still reversible methods.

So we urge you: be appropriately wary of repeated risks and what you can do to mitigate those risks! There are many things that are fine to do once, but that are dangerous to do T times, for a moderately large T.

*[A semi-technical side note for the mathematically inclined: when we do the activity twice, the risk of the bad thing happening the first time OR the second time if each event is independent is actually R + R - R^2. The R^2 comes about from considering the possibility that the bad thing happens both of the times. For intuition on this point, consider two circles that are slightly overlapping. To find the total area of the circles (i.e. the total probability of the events represented by the circles) you can't just add up the two areas (i.e. add the probabilities), because you'll double count the overlapping part (i.e. the part representing the event occurring in both instances), so you'll have to subtract something away (the R^2 in this case, which has this form due to the assumption that the events are independent). However, if R is small, then R^2 is super super small, so we can ignore it by assuming it's 0 — hence the total risk ends up being about 2*R = R + R - 0. Therefore, the math mentioned above breaks down if you're doing a thing with really high risk, or if you do the thing a large number of times relative to the risk level — this makes sense because if you have a 50% chance of dying each time, then clearly if you do it 4 times you don't have a 4*50% = 200% chance of dying, and likewise, if you have a 1% chance of dying each time, and you do it 200 times you also don't end up with 1%*200 = 200% chance of dying. But for small, independent risks that you do a moderate number of times, R * T is a good rule of thumb.]

**[Note: sometimes the risk PER time that we do something goes down the more we do it, because we become more skilled. For instance, if you swim in the ocean as a weak swimmer, you are more likely to drown than if you do so as a strong swimmer. But then again, as a strong swimmer you may take greater risks — such as by swimming on stormy days — and as a weak swimmer, you may be cautious and stay very close to shore, so the risk per time on average could increase as you do it more. It's hard to make a general rule about how the risk per time varies as you do something more often.]

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