There have been many, many people on the Internet and elsewhere who have talked about the “many” lives saved by the quick actions of several layers of responders this week to close the Key Bridge to traffic. I’ll pick on Yonah Freemark here (sorry, Yonah), but you can find any number of twitterers talking about how many lives were saved. This guy said “it could have been hundreds” in an otherwise informative interview. The New York Times story about the audio says it “likely saved lives” which, as I’ll explain, is probably better phrasing.

The bridge collapse occurred at 1:30 in the morning. There’s a pretty good chance that, had the mayday call not gone out and had the local authorities not intervened, there would have only been two vehicles on the bridge, or perhaps none. The presence of the construction crew, which was not able to evacuate the bridge, may have been the reason the response was so quick, since there were police vehicles nearby. This leads to two lessons and a thought:

• Major bridges over busy shipping channels should have means to quickly stop traffic (perhaps even automated gates)
• Construction crews on such bridges should have good communications and be prepared to quickly evacuate.

The thought is that people vastly overestimate the use of highways at very off-peak times of day. The MTA probably has better data from electronic tolling, but the most recent public traffic count for this bridge seems to be from 2019 (the user interface for Maryland’s traffic counts—like most such interfaces—leaves a lot to be desired). It includes an early Tuesday morning, so it’s probably good enough for a rough estimate. From that, we can find that somewhere around 180 vehicles cross the bridge per hour between 1 and 2 a.m., which would mean that at any given time, there would be approximately 3 vehicles on any given mile of roadway. Since the collapsed portion of the bridge is about 2/3 of a mile long, we would expect there to have been two vehicles on the bridge when it collapsed, right?

Well, kind of. We actually have tools for figuring this out. What’s that music? It … sounds like the entrance music for the Poisson Distribution! The Poisson Distribution is “a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time if these events occur with a known constant mean rate and independently of the time since the last event.” This is basically how traffic works. (It’s also, for example, how passengers arrive at a transit node.)

We can use the Poisson distribution to find the probability that n vehicles would have been on the collapsed portion of the bridge given the traffic levels. This gives the following:

n = 0: 14%
1: 27%
2: 27%
3: 18%
4: 9%
5: 4%
6: 1%
7 or more: 0.4%

So the modal outcome is (1 or 2) but there’s about the same chance of no cars on the bridge as four or more at that time of day. Given that vehicle occupancy overnight is likely not much more than 1, the number of lives saved by this specific chain of actions is likely in the range of zero to four. If you watch the video, you can see the number of cars on the bridge basically ranges between 0 and 3. As we’d expect. Statistics are fun!

That said, there’s a lot this leaves out. Additional vehicles on a bridge approach could have Thelma-and-Louised off the bridge without warning given driver inattentiveness (although “road” is a major detail). More importantly, having these policies in place means that had this occurred at a higher traffic time of day, it would have prevented a mass casualty event (although it seems like the presence of the work crew may have been part of the reason there were police officials nearby, so we may be very lucky this occurred at 1:30 a.m. and not 5:30 p.m.). Using the Poisson and the peak traffic of 2700 vehicles per hour, there would be a 96% chance of at least 20 vehicles on the span, a 45% chance of at least 30, and a 3% chance of at least 40. (Still, “Hundreds?” Well, Poisson does have a long tail but the probability of that would be quite small.)

Beyond that, there’s an outlier event where the bridge has a traffic jam and vehicles couldn’t clear the bridge with advanced notice. This is why, for example, in the Mont Blanc tunnel, which had a previous mass casualty event, has very specific rules about vehicle separation, and the Tobin Bridge in Boston is closed when an LNG tanker passes underneath. But this roadway had a low-enough traffic count and is far enough from bottlenecks that it’s likely very rare that traffic ever backed up onto the bridge.

There’s also an argument that the ROI for rebuilding the bridge isn’t there for 32,000 vehicles per day (again, thanks, Yonah). Some caveats there. 1) The bridge had $57 million in toll revenue in 2023 ($31 million cars, $26 million commercial vehicles). This is about 8% of the state’s toll revenues, and much of the traffic might move to other tolled facilities. But there would be an overall loss of revenue. 2) The bridge is a haz-mat route and with the other two routes through Baltimore in tunnels, haz-mat would have to detour around the Baltimore Beltway. 3) Would the State of Maryland be able to recover salvage damages from the ship owners and insurers if they chose not to rebuild the bridge? I don’t know, I’m not a lawyer. (Or, for that mattter, a bridge engineer.)

Will such a boat-on-bridge failure happen again? There’s no reason to think it won’t. Ships are getting bigger, and there are only a couple of dozen bridges of this magnitude over waterways with large ships in the country (San Diego, San Francisco and Portland on the West Coast; Houston, New Orleans and Tampa on the Gulf; Jacksonville, Charleston, Chesapeake Bay, Philadelphia, New York and Rhode Island on the Atlantic, plus potentially Boston, Portsmouth, the Penobscot and a couple of canals, although these have much less traffic). In a sense, it already has, and the new Sunshine Skyway has significant deflection infrastructure around it (we just didn’t learn from that disaster). Perhaps this will be a wake-up call to reinforce deflection around vulnerable bridge supports. In the shorter term, large bridge operators should consider having SOP both for bridge users and bridge workers in case of this sort of event.