Probability, camps, and COVID

 There's been a fair bit of discussion about people running summer camps this summer. Right now, I've seen both Christians and Jews arguing that their summer camps for kids should run, and that with appropriate safeguards there won't be any serious risk of issue. There's a lot wrong with this argument, but one aspect which is possibly not getting enough attention is something that is an implied assumption people are making without even realizing it, which is that if one increases the scale, the number of probable infections will rise linearly and that the chance of an infection event rises roughly linearly. Neither of these are the case. My suspicion is that this sort of assumption is occuring not just in the camp discussions but in all the discussions about opening up. So let's discuss these ideas. 

 

First, let's imagine a game. In the first game, you'll get to roll a regular six sided. If it turns up 1, then one loses the game, otherwise one wins.  The chance of losing is 1/6. Now, play the same game, but roll 10 dice. The chance that at least one  show up is much much higher. If a one represents an infected person, then as one increases the number of people, the chance that at least one infection is present goes up drastically. This is, by the way, part of the point of isolating in small groups, rather than just isolating with a larger number all of whom seem safe. (And yes, the chance that anyone is infected with COVID in life is less than 1/6 but the same basic pattern holds.)

 

Now, we're going to imagine a different game. In this game, there are a whole bunch of people. When we start one person has a red slip of paper, and everyone else has a blue slip of paper. We'll play the game in stages. At each stage, every person shakes hands with every other person,  but if one of the people they shake hands with has a red slip of paper then they roll a fair 10 sided die, and on roll of 1, they give a red slip of paper to the other person. Having a red slip here represents being infected. For simplicity, we'll imagine that only two rounds happen, so first everyone shakes hands with everyone else, paper is given as necessary,  and then a second round of handshakes occur. 

 

Now, we can ask, how likely is anyone to have a piece of red paper at the end? Let's look at the simplest case, where we have two people playing, one with a blue slip and one with a red slip. The chance that the person with the blue slip is 19%. To see this, note that  there's a 10% chance in either round that they'll get a red slip , but that double counts the possibility they get a red slip in both rounds, which has  1% chance  (1/10 times 1/10 of happening). Now, let's imagine that we play this game with 3 people; a little figuring will convince you that the chance now that either of the two people with blue slips end up with a red slip has gone up: why? Because in the second round, there are more pathways for infection, if one of the two with blue was infected on the first round. By the time one gets to 20 people playing this game,  most people with blue slips will end up with a red slip by the end. If one plays with  30 people, almost every blue slipped person will end up with a red slip.  

 

Shaking hands here is of course a proxy not just for shaking hands but for many other forms of contact,  including coughing and  touching. In the case of teenagers at camps there is probably more other experimental behavior than some of the parents are comfortable thinking about even if they remember their own time at such camps.  But the essential upshot is the same; if one has more pathways, the chance for infection goes up drastically.  This really is not just about summer camps; similar remarks apply to any gathering of people or extended way for people to interact.