By Bob Rietz

I’ve flown about once a month for 25 years and it finally happened. For the first time I can recall, I ran into someone I know in an airport. What’s the probability of that?

However, it was actually not the first time this has happened. I’ve met colleagues traveling to or from the same actuarial meeting that I was attending—clearly not a coincidence. I’ve also met neighbors in the Asheville airport, with both of us on our way to Atlanta. where our paths would diverge. Bumping into an Asheville friend who is waiting for the same aircraft to return home from Atlanta doesn’t seem unusual.

All my flights currently include a transfer in Atlanta, and I’m puzzling over the probability of encountering a random acquaintance there. Let’s narrow the question to the probability of bumping into them in the Atlanta airport on any single trip, but excluding fellow Ashevillians. I lived in the Detroit area for 35 years and often fly there, but I’ll include meeting a fellow Detroiter by chance.

This analysis requires *way* too many assumptions to list (the editor restricts me to one page), so I’ll mention only the primary ones. Assume that I know 1,000 people. But the probabilities of meeting any one of them in an airport are not the same. I’m much more likely to meet another actuary, former client, or prior work colleague during my travels than I am to meet any of my grandchildren who hardly ever fly. Another very significant assumption would be the distribution of how frequently these 1,000 people fly each year. With a giant leap of faith, I’ll assume that half of them (500) fly somewhat regularly, and of that half, they are evenly split between flying more frequently and less frequently than I do. Next I’ll assume, on balance, that these 500 people fly as frequently as I do. This is admittedly a weak assumption, but it greatly simplifies the subsequent math.

I fly almost exclusively on one airline (call it Alpha), and some of these 500 acquaintances are frequent flyers on other airlines. So, another important assumption is how many of these 500 acquaintances also fly on Alpha. Bear with me as I attempt to get to this number.

About 860 million people flew in the U.S. in 2017, and Alpha flew around 150 million of them. I’ll divide these two statistics—150/860 = 17.4%—to assume that 87 of my 500 acquaintances who fly, do so on Alpha.

Alpha has several hubs, and these 87 acquaintances who also fly on Alpha might be flying through another of those hubs. Atlanta enplanes and deplanes about 100 million passengers a year, but most of them are being double-counted. So, let’s assume that 50/150 of my 87 acquaintances (now 29) who fly Alpha will fly through Atlanta about once a month. Analysis of these 29 people requires combinatorial probabilities, which are a little hazy (I passed that exam 50 years ago), but here goes. And I welcome any reader who corrects these calculations.

Start with the day I leave Asheville. A month has 30 days, so the probability of my flying through Atlanta on the same day as at least one of my 29 remaining acquaintances is 62.6%. Atlanta has flights 18 hours a day, so the probability that we’re both there within an hour on the same day is 3.5%. I invariably must transfer to another of its seven terminals, as will any of my acquaintances who are flying that day, which lowers the probability of meeting them to 1%. I have the same probability of meeting them on the day I return to Asheville, which raises the probability of meeting someone in any one month to 2%.

Thus, the probability that I *won’t* run into someone I know during one of my monthly flights is 98%. The probability I’ve *never* met an acquaintance over the last 25 years is that probability to the 300^{th} power, which is 0.24%. So, the probability of bumping into someone I know in Atlanta over a 25-year period is a near certainty. This suggests that either I’ve wandered through airports oblivious to seeing anyone I know, or my memory is getting worse.

How many times have you run into someone you know in an airport?

** BOB RIETZ is a retired pension actuary who lives in Asheville, N.C.**