It is summer time here in Australia and hence I find myself at the beach quite a bit. So naturally I want to talk about gruesomely dying in the jaws of a shark. Biologists often claim that the risk of dying from a shark attack is so inconsequentially low that any rational person would ignore it, in comparison to the many risks we take doing mundane activities like driving or taking selfies. Often the statistics quoted go something like this

Number of shark attack deaths pear year: 1

Number of car accident deaths per year: 38,300*

This indeed says that deaths from shark attacks are incredibly rare, but it says absolutely nothing about the relative risk of dying from a shark vs. a car. The numbers are meaningless without an appropriate denominator (that pesky number at the bottom of a fraction). The denominator here is “years”, as the statistic is “deaths per year”, but is that the correct choice for identifying the risk of death when choosing between activities? I don’t think so. There are many people who never venture into the ocean, and of those who do, most visit only a few times per year. In comparison, the average person in the US drives nearly every day. In other words, how many times do people really have the opportunity to encounter a shark?

So below I calculate a more meaningful statistic, the probability of death per instance of exposure (or at least a very rough estimate). Doing so, we can determine the distance one would have to drive in order to obtain the same chance of dying as someone going to the beach and dying from a shark attack. It starts with the numbers below

exposure | source | |

Beach visits / year in USA | 110 million (1.1 x 10^{8}) |
National Oceanic Atmospheric Administration |

Miles driven / year in USA | 3.1 trillion (3.1 x 10^{12}) |
US Department of Transportation |

The risk of dying from a shark attack in a given beach visit is therefore roughly 1 in 110 million and the probability of dying per mile driven is approximately 38,300 in 3.1 trillion (or roughly 1 in 81 million). What does this mean? These numbers are quite close, the risk of death from driving 0.74 miles (or 1.2 km) is about as high as dying from a shark during a beach visit.*

Now you can look at these numbers and think, the risk of dying from a shark attack is so low … it is equivalent to less than a mile (or a little over one km) of driving. Alternatively, you can look at these numbers and say wow … the statistic, “1 death from a shark attack vs. 38,300 deaths from car accidents” really makes the risk of dying from sharks sound a lot more inconsequential than the calculations above. Which camp you find yourself in might depend on how much you drive or visit the ocean without using a car. I’m gladly happy to visit the beach and take such a small risk, completely ignoring the chance of being eaten by a shark, but perhaps the risk isn’t as inconsequential as I once thought. Whatever your thoughts, the reminder here is that it is important to think about the appropriate denominator when talking statistics (there is almost always some assumed denominator, whether we realize it or not … absolute numbers are often misleading).

*These calculations required some assumptions. First we assumed the numbers from the above sources were true. We also assumed that everyone at the beach goes in the water, which likely isn’t true – the risk of dying due to a shark attack might be more like the risk of driving one or two miles if for example only half of beach goes ever go past ankle-deep in the ocean. We also assumed that shark attacks and auto-accidents occur at a fixed rate for all individuals. This is of course untrue, by driving safely or taking safety precausions in the ocean you can reduce your risk of dying in either situation. We are merely looking at averages here.

Nice follow up on my mumbling from the lab meeting!

Would it be good to use “hours in the water” and “hours in the car” as a more approriate measure for comparison?

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Yes that would be ideal! If only we had data on total hours spent by people in the ocean. That would be an even more appropriate denominator.

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But would you trust the numbers given by beach users? I reckon every surfer has told their boss/partner they were “only in the water for an hour before work this morning…”

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@scott. This is a good point. It is indeed possible that the number of beach visits is severly underreported. I believe that NOAA’s methodology is to randomly survey people at several selected beaches and then use a model to extrapolate the number to the whole US. If beach visits are under-reported, the risk of a fatal shark attack could be much lower than I calculate here (and the number reported here is already extremely low, 1 in 110,000,000 beach visits).

I really want to emphasize again that the risk reported here is extremely low, I can’t say this enough. But the risk of getting in a fatal car accident, if you drive one time, less than a mile, is extremely low as well.

If the surfer in your example drives to work, (the average commute is probably around 10 miles), that means even if a surfer goes to the beach every single day, she is over 25 times more likely to die in a car accident than in the jaws of a shark (and that’s someone who surfs everyday!).

I had an interesting conversation with a friend who said that this post might get misinterpereted in the opposite direction, where people read this and start thinking shark attacks are common. My friend pointed out that there is this commonly thrown out statistic that 50% of all car accidents happen within 5 miles of home. While this is true this is a rediculously misleading statistic (probably more so than the raw numbers of shark attacks, all it really says is you can get in an accident anywhere, and that obviously people who drive spend a lot of that time within 5 miles of their home). But because people translate this into “I have a high probability of getting into a fatal crash if I drive half a mile to my friend down the street”, which is flat out false … perhaps the narative I tell above is actually worse than reporting the raw numbers.

Context really matters … something we mathematicians often forget. I hope my post doesn’t get taken as “the chance of getting fatally attacked by a shark is high.” That is definitely wrong.

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So if you drive to the beach for a swim do you double your chances of dying? 🙂

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Depends on how long you drive. If you drive 0.6 km to the beach (each way) you double your chance of dying. If you drive 60km to the beach your chance of dying goes up by a factor of 100! Probably best if we walk to the beach :).

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You cannot determine if it is safer to walk to the beach until you do the pedestrian death statistics.

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Good point! Let’s look at the stats, 42 billion miles worth of walking trips in the USA*, and there are approximately 5,376 fatalities. So 1.28 x 10^-7 deaths per mile walked. This means that walking 0.07 miles has the same risk of death as a shark attack from one beach visit. So your house would have to 60 meters away from the beach or less for the walk to be less dangerous. In other words, the walk from your car to the beach may often be more dangerous than dying from a shark attack.

*based on the figure combined with the last table in this source http://www.pedbikeinfo.org/data/factsheet_crash.cfm

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And consider how your risk goes up when you are surfing at an Australia beach frequented by sharks. There is no comfort in the general death statistic once you put yourself in that situation. Same with death stats for any risky activity. Once you engage in the activity (ice climbing, heli-skiing, sky-diving, abalone diving, Russian Roulette), you are at far greater risk of death than the average person is from that activity, since the average person doesn’t engage in that activity. Death by cannibal is a one-in-a-billion risk until you’re marooned on an island of cannibals.

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It is almost certainly true that the average surfer is far more likely to die from a shark attack than a general beach goer. One blogger attempted to, roughly, calculate this risk and estimated it to be three times more likely than dying from a lightening strike. Still extremely unlikely, but indeed higher than the stats here suggest. It should be noted that their calculations are likely to overestimate the risk because they assumed all shark attack deaths happened to surfers. http://www.surfer.com/features/what-are-the-odds/#qFt5uCUYO4ecIFIe.97

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