Recently I heard that there had been three simultaneous emergency service calls on one night, straining the capacity of our EMS that is staffed in part with volunteers. This got me thinking: given our population now, what are the odds of three simultaneous emergency service calls? And, perhaps more importantly, how fast would this probability increase as our population increases? The more often multiple emergencies occur at the same time, the more the capacity of fire and EMS needs to increase in order to avoid delays in response to any one of those incidents. It turns out we go from a state of a low probability of simultaneous incidents to one of high probability faster than our population grows.
Investigating this question requires a foray into the world of probability. To simplify, a good first step is to recall a standard exercise in most intro to probability courses: the birthday problem. The basic problem is, given a group of people at a party, what is the probability at least two of them have the same birthday? Assume leap years are excluded and all 365 days are equally likely. I had to look back at a lecture with the answer that I saw in a MOOC, Harvard’s Statistics 110, taught by Joe Blitzstein.
By the pigeon hole principal the probability reaches 100% when you have 365 people. Most people are surprised to learn, however, that there’s a greater than 50% probability with only 23 people in a room, and the probability reaches 99.9% with only 70 people in the room.
The formula to find the probability of three birthdays falling on the same day is more complicated, but there’s a way of approximating a solution using a probability distribution known as the Poison distribution. With 30 people in a room, there’s approximately a 3% probability of three birthdays occurring on the same day. With 100 people, the probability is over 70%. So as the “population” in the room roughly triples, a situation analogous to Teton County over the past 30 years, the probability of three random birthdays falling on the same day goes up by more than 20 times. At some point Teton County, by the pigeon hole principle, will have a 100% probability of having multiple triple-call scenarios. This level of population might be well into our future. Note, however, that the probability of simultaneous three-call scenarios grows faster than our population grows until we are over a 95% probability.
So far we haven’t answered the question, “What’s the probability in Teton County of three simultaneous emergency calls?” That’s a far more difficult one, especially given how our population fluctuates during the year. However this does point out that we should expect our need for fire and EMS services to grow faster than our population. How is the subject of future blogs.