At a Science Café talk about mammals tonight, I heard this statement: "even if the chance of it happening is one in one-million in a given year, the likelihood is high given a span of one-million years." Is that really true? What about in the general case of
"if the chance of an event happening is 1 in n, what's the likelihood that the event will occur at least once for n number of trials?"
The expression for that probability is 1 minus the chance that the event will not occur at all during the n number of trials. The chance of the event not occurring each time is (n-1)/n. Raise that to the n-th power for the number of iterations, and subtract from 1, which represents certainty, and the probability that the event happens at least once is:
We wish to find the limit as n approaches infinity. Working out the L'Hopital's rule by hand can be somewhat cumbersome with the exponential term, but with the help of Mathematica, we determine the limit to be:
The answer approximates to = 0.632120559. As n increases, the output probability decreases. However, n only needs to be greater than 485 for the output probability to match the first 3 decimal places after rounding. Of course for any given probability, as the number of trials increase indefinitely, the overall probability of the event occurring at least once approaches certainty. But in the special circumstance here, where the number of trials (n) match the inverse of the probability of each event (1/n), the overall probability approaches (e-1)/e.
