# How Are Sharks Less Dangerous than Vending Machines? An Exercise in Conditional Risk

(iStockphoto)

Did you know that vending machines, not a major danger in most of our minds, are twice as likely to kill you as a shark? I heard this statistic at the new shark-and-ray touch tank of the New England Aquarium, which I try to visit weekly with my daughters. You stand at a large, shallow tank with plexiglass walls and can lay your hand in the water, gently feeling the sharks and sting rays swimming by.

The aquarium probably wants to convince visitors that sharks are not the fierce predators of Jaws fame, and thereby help protect sharks from hunting and extinction. Although I could admire this motive, the comparison always surprised me. My number sense complained that sharks simply must be more dangerous than vending machines.

However, upon looking up the risks, I found that the comparison was correct. The yearly risk (in the United States) of dying from a shark attack is roughly 1 in 250 million. In contrast, the yearly risk of dying from a vending machine accident is roughly 1 in 112 million. The vending machine is indeed roughly twice as lethal as the shark!

Why then was I still troubled by the comparison? Maybe my number sense needed a tune up, and I should just accept the statistical facts of life. I then started thinking about it using the method of easy cases. This method, along with proportional reasoning (the tool in this post about colonial-era literacy), is one of my favorite tools for developing what I like to call number-sight: the ability to see connections among (and make sense of) the myriad numbers around us.

The easiest case is often an extreme one. My own extreme case of shark-attack risk happened while teaching at the African Institute for Mathematical Sciences, in Cape Town, South Africa. The institute is right on the beach, so one day I tried learning to surf (with more emphasis on “try” than on “learn”). I soon heard whistles from the lifeguards. Because the water was packed with swimmers, I assumed that swimmers were going too far out. That’s what the whistles meant on the New Jersey beaches in my childhood. As an adult who knew how to swim, why worry? After returning to shore, I learned that the whistle was warning everyone of a great white shark that had been sighted swimming around the bay. It was probably the same shark that had bitten the leg off a surfer a few months earlier.

Don’t tell me that, while surfing in that bay on that day, a vending machine posed more danger to me than that great white shark! From this extreme case, I realized the problem with the comparison. These statistics are averaged over everyone in the United States. In most places in the U.S., such as Kansas, people are nowhere near a body of water with sharks. The comparison of the risk to a vending machine, while true as far as it goes, ignores highly relevant information—such as whether one is swimming in the same bay as a shark.

The comparison also ignores important information about vending machines. After all, how do people die from a vending machine? Vending machines are not known carcinogens. I imagine that the machine takes someone’s money and malfunctions. The customer then shakes it to free the snack, whereupon the machine tips over and crushes the hot-tempered purchaser. As the doctors say, “Don’t do that then!” Keeping cool in this difficult situation probably reduces the vending-machine death risk to zero.

This problem of implicit but essential statistical information is wonderfully illustrated in this XKCD cartoon:

#### BrianM

In mortality there is a concept called 'exposed to risk'. As the article says, the answer is not to compare the deaths with the whole population but those who have some exposure. For mortality this is usually straightforward as you have some sort of population. In other cases defining the exposure is not always easy, as the comments show - measure air travel by miles travelled or time spent travelling? And then measuring it can be even harder. The real danger is resorting to what can be measured, in this case the whole population, rather than measuring the true exposure. In one sense the statistic is right - pick a random person in the USA and they are more likely to die from a vending machine than a shark. But measure those who use vending machines v those who swim in the sea and you would probably get a different answer.

Despite that this does not remove the basic point - the risk from sharks is truly limited. One death per annum per 250 million is not very much.

#### chris

It's definitely the perceived threat and the nature of the death. I dive with sharks every week and take people to see them. Often they are apprehensive of the shark before the dive but afterwards they find themselves enjoying the experience. The thing is we are in an environment that is unnatural to us and we don't feel in control. Put into that mix a slight possibility that a large animal with lots of teeth may attack us and we get scared. Don't get me wrong sharks are not harmless but the amount of human / shark interactions that don't end in attack are staggeringly huge. They just don't make for good TV. I know that where I dive millions of people over a year go into the water and I've seen how many sharks are close by. (Great whites are spotted every other week) and there has been one attack in 20 years.

#### Lauren

Actually your undersatanding of the stats of shark attacks and vending machines is off. People putting their arm in a vending machine and it falling on them is causing the accident to take place and is counted. If you put your arm near a shark and it bites it is provoking a shark to bite and is not counted. You do have a risk of being in a unprovoked shark attack, but there is no chance of being involved in a un provoked vending machine accident. Keep beleiving those US folks and be a fool when it comes to sharks.

#### bill

Isn't this a similar example as the dangers children drowning in a in a swimming pool or being killed by a gun which appeared in Freakonomics?

#### Rags Srinivasan

Isn't the use of probability here follows the frequentist approach - counting all possible instances and declaring the chances of next episode of shark attack or vending machine death?
When you add the conditional probability clause (that you allude to with your surfing story) then the frequentist estimate gains some refinement and start to represent an estimate of the likelihood of the attack.
P(Killed by shark | you never ever venture into shark infested water) is indeed much smaller than
P(Killed by vending machine | you go by vending machines once in a while)

but
P(Killed by shark| you are in the water where shark had attacked just recently) is way more than the other case.

-rags

#### Eric Stott

What I don't really understand, and I guess that I am economics stupid:
What do any of these numbers mean?
What is the risk for me who never swims in an ocean (I live in AZ), and only walk by vending machines?
If I am only swimming in dry suits in 36 degree ocean water, aren't I pretty much guaranteed to be free from sharks?
What is the probability of a vending machine up and tipping over on me as I walk by?

These statistics make more sense to me. Does everyone who goes to the vending machine shake it so violently that it 'could' tip over?
Does everyone swim in shark infested waters with a huge gash on their leg, bleeding like crazy?

This makes more sense than simply scaring everyone, me included! When I was a kid we took a trip to California, and while swimming there, the only thing I could think about was how many sharks were out there nipping at my toes!

#### RPaulH

http://www.funfacts.com.au/vending-machine-death/
darwin at work - in the land of Darwin (australia)

#### RPaulH

http://www.darwinawards.com/darwin/darwin2001-25.html

likely that alcohol is involved in many of these deaths

35 died in 20 years according to this link/report

#### Ben Maral

The comparison between shark attack risk and vending machine risk is nonsense.

Unless you enter shark occupied waters the chances of a shark attack is obviously zero.

The comparison should be between swimmers in shark infested waters who use vending machines and swimmers in shark infested waters who do not use vending machines.

#### Kayla E

I dislike the fact that advocates for the sharks are using the argument that sharks aren't as dangerous as we all feel like they are as a primary reason why we should care about their preservation.

#### Craig

Clearly the author has never seen a little documentary called "Sharknado."