Perhaps like you, I wonder from time to time what my chances of dying on an airplane are. These thoughts typically occur when I am actually on an airplane, but, earlier today, I was fortunate enough to have this thought within arms reach of my computer and the internet. A quick google search told me the probability of dying in any given flight is roughly 1 in 11 million. If we accept this number, and the assumptions that come with it, then it is a short calculation to get the odds of dying in N flights.
The mathematics of this problem is just a little bit counterintuitive. Namely, if the probability of dying after a single flight is p, the odds of dying after two flights is NOT 2*p. Instead, it is p+(1-p)*p. The first term is the probability of dying in your first flight and the second term is the probability of dying in your second flight. The subtle part is that in order to die on your second flight, you must not die on your first flight, hence, the factor of (1-p) in the second term. In addition to being logically correct, this small detail is what prevents p(N) from exceeding 1 for arbitrarily high N.
The general formula for the probability to die within N flights is the sum of dying in each flight leading up to the Nth flight:
The mathematics of this problem is just a little bit counterintuitive. Namely, if the probability of dying after a single flight is p, the odds of dying after two flights is NOT 2*p. Instead, it is p+(1-p)*p. The first term is the probability of dying in your first flight and the second term is the probability of dying in your second flight. The subtle part is that in order to die on your second flight, you must not die on your first flight, hence, the factor of (1-p) in the second term. In addition to being logically correct, this small detail is what prevents p(N) from exceeding 1 for arbitrarily high N.
The general formula for the probability to die within N flights is the sum of dying in each flight leading up to the Nth flight:
where s is the probability of dying on a single flight, which we took to be 1/(11,000,000). Since the sum can be written in closed form, we have a closed form equation for the probability of dying in N flights. This function is plotted below.
Thus, we see that in order to have something like a 50/50 chance of dying on a plane during your life, you would have to fly about 7.5 millions times! If you take 4 flights a day it would still take you 5000 years to complete that many flights. So, I guess the point here (if there is a point?) is that you are most likely safe on a plane, regardless of how many times you fly. Of course, your chance of death goes up relative to someone who never flies, but it remains extremely improbable.