Here’s a simple probability problem that should be accessible to high school students who have learned the Multiplication Rule:
Suppose that you play the lottery every day for about 20 years. Each time you play, the chance that you win is chance in . What is the probability that, after playing times, you never win?
This is a straightforward application of the Multiplication Rule from probability. The chance of not winning on any one play is . Therefore, the chance of not winning consecutive times is , which we can approximate with a calculator.
Well, that was easy enough. Now, just for the fun of it, let’s find the reciprocal of this answer.
Hmmm. Two point seven one. Where have I seen that before? Hmmm… Nah, it couldn’t be that.
What if we changed the number in the above problem to ? Then the probability would be .
There’s no denying it now… it looks like the reciprocal is approximately , so that the probability of never winning for both problems is approximately .
Why is this happening? I offer a thought bubble if you’d like to think about this before proceeding to the answer.
In particular, for the special case when , we find
The first limit can be proved using L’Hopital’s Rule. By continuity of the function , we have
The right-hand side has the form as , and so we may use L’Hopital’s rule, differentiating both the numerator and the denominator with respect to .
Applying the exponential function to both sides, we conclude that
In an undergraduate probability class, the problem can be viewed as a special case of a Poisson distribution approximating a binomial distribution if there’s a large number of trials and a small probability of success.
The above calculation also justifies (in Algebra II and Precalculus) how the formula for continuous compound interest can be derived from the formula for discrete compound interest
All this to say, Euler knew what he was doing when he decided that was so important that it deserved to be named.