Unexpectedly Intriguing!
March 12, 2020

If you're going to a social event with lots of people, what is your risk of coming into contact with someone who has been infected by the coronavirus?

Alex Tabarrok posted the math that can provide insight into the answer to that question, which we'll convert into a tool you can use to run your own scenarios. Here's how he describes the math:

The mathematics for calculating the probability of exposure given the number of carriers in a population and group size aren’t difficult but they can be surprising. Even a low number of carriers can generate a relatively high probability for reasonably sized groups. For example, assume you run a firm of 1000 people in the San Francisco Bay Area (population 8 million.) Let’s suppose that there are just 500 carriers in the area. In this case, assuming random draws, the probability that at least one of your employees is a carrier is 6%. You can run your own calculations at Wolfram Alpha following this format:

p=8000000, c=500, g=1000, 1-1(1-c/p)^g //N

where p is the population size, c is the number of carriers, g is the group size and the //N at the end isn’t a division but a command to Wolfram Alpha to give you a numerical answer.

We're huge fans of Wolfram Alpha, but since the data entry is a bit cumbersome, you'll hopefully find the user interface in the following tool to be a bit more friendly. If you're accessing this article on a site that republishes our RSS news feed, please click through to our site to access a working version.

Data for Population, Group, and Number of Infected
Input Data Values
Overall Population
Number of Infected (Carriers) Within the Overall Population
Size of a Group (Subset of the Overall Population)

Probability of Carrier Within The Group
Calculated Results Values
Probability of a Carrier Being Within the Group

One important thing to note is that the probability the tool calculates assumes that carriers are randomly distributed among the whole population. In practice, many will be concentrated within smaller groups of the population, where effective quarantine practices, or what the kids are calling "social distancing" these days, will reduce the probability of exposure.

Alex makes the key point for what we can learn from the math if you're thinking of going to a public event while the risk of viral infection remains high:

Now here is the most important point. It’s the size of the group, not the number of carriers that most drives the result. For example, suppose our estimate of the number of carriers if off by a factor of 10–that is instead of 20,000 there are just 2000 carriers in the United States. In this case, the probability of at least one carrier at a big event of 100,000 drops not by a factor of ten but just to 45%. In other words, large events are a bad idea even in scenarios with just a small number of carriers.

You can confirm that insight using the tool above, or if you prefer, here is Joshua Weitz' COVID-19 Event Risk Assessment Planner that presents the data visually:

Joshua Weitz' COVID-19 Event Risk Assessment Planner Chart

Numbers like these are why the NBA has suspended its season.

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