Bill Gates retweeted a World Health Organization infographic showing that mosquitos kill vastly more people than sharks every year—on the order of 100,000 times more.

In his tweet Bill captioned the infographic with, "Why I would rather encounter a shark in the wild rather than a mosquito." Presumably he’s referring to man-eating sharks.

This was an informal comment designed to highlight the misery caused by malaria—a cause that is at the centre of Bill’s philanthropy. Clearly it wasn’t supposed to be a serious risk assessment.

But it illustrates how confusing conditional probabilities are, and how easy it is to make invalid statistic inferences.

The data in the infographic refer to the probability that, given you are dead, you were killed by a shark or a mosquito. Chances are that it was a mosquito—not a shark. That seems intuitive.

Technically, we can denote this as

$P(shark|death) << P(mosquito|death)$

I’m not convinced by Bill’s implication that it’s better to encounter a shark than a mosquito. I grew up after "Jaws" was released. Intuitively, surely sharks are much more dangerous, right?

The risk posed by meeting either of these creatures is the probability of being killed given you met them. If we encountered man-eating sharks as often as we encounter mosquitos we’d be getting munched on constantly.

Sharks are definitely scarier. We can represent this more formally as

$P(death|shark) >> P(death|mosquito)$

It’s important that we distinguish between $P(mosquito|death)$ and $P(death|mosquito)$ when drawing inferences.

Fortunately man-eating sharks live in the ocean and I don’t. Given that, I’m willing to sign up for more killer sharks and less mosquitos.