Small probabilities add up

Rebecca Watson made an excellent video about how the recently-released COVID-19 risk map is not particularly helpful, due to the lack of context provided and the public’s general lack of understanding about how statistics work. It’s well worth watching (and also talks about a few other things, like issues with services like 23andme’s genetic risk factor screening kits). People are apparently using this risk map as a means of justifying going to Thanksgiving gatherings based on “only a 5% infection chance” or the like, which is incredibly short-sighted.

Let’s say that 5% of the population is carrying the disease, and assume that this statistical model is completely accurate. (It almost certainly isn’t, but that’s beside the point.) This means that any time you encounter someone there’s a 5% chance that they’re infected. That seems pretty low, right? For a single encounter, sure. But that doesn’t tell the whole story.

Probabilities are a bit counterintuitive; you can’t simply add them together to work out what the overall probability of something is and instead you need to think about what you’re actually trying to measure. When you’re considering the risk of being exposed to COVID-19, what you’re really hoping for is the probability that you won’t be exposed. So, given a 5% exposure risk, it’s pretty easy to tell that you really have a 95% chance of not being exposed. That sounds great, right?

Well, that chance is on a per-person basis, and you combine positive chances by multiplying them together. Every single time you are exposed to another person, that’s another 95% chance that you’ve still not been exposed. So, if you’re exposed to N people, your chances of not being exposed are 0.95N. (Reminder that % is just shorthand for “per 100,” i.e. “this number divided by 100.” So 95% and 0.95 are the same number.)

If you’re at a Thanksgiving celebration with 10 other people, your not-being-exposed chance is 0.9510 = .599 — a 60% chance you’ve not been exposed, or a 40% chance you have been.

And remember that COVID-19 can linger around due to airborne and surface-borne transmission. Going to the grocery store in a “5% risk” area? Let’s assume that you’re encountering around 100 peoples' worth of exposure risk. That means that your COVID-19-negative chances are .95100 = 0.0059 — there’s a 99.4% chance you’ve been exposed to the virus! You’d better hope that everyone else was wearing a properly-fitted mask and practicing good hygiene.

Okay, so, the risk map accounts for the number of people at an event. But the risk doesn’t reset every day. At best you can consider your exposure window to be every person you’ve come across (and whatever lingering residue they’ve left behind) over the past two weeks. Think of all of the people you’ve been in the same room as during that time, and all the people they’ve been in the same room as, too. What are the chances any of them have had COVID-19? Pretty darn good.

Simply knowing the overall infection rates doesn’t tell you a whole lot about your chances of being infected. You need to know how many people you’re encountering and what the infection transmission rate is like. Indoor gatherings with food are the most effective way of transmitting a disease. Outdoor spaces where everyone is wearing a mask and avoiding direct contact are the safest. But even small probabilities add up.

Wear a mask, wash your hands, and maintain a safe distance.

And stay at home for Thanksgiving.


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