Suppose I have some random process described by a binomial distribution, with "success" probability $p$. For $n$ trials, the expected number of "successes" obeys

$$\begin{aligned} P(k) &= {n\choose k} p^k (1-p)^{n-k} \end{aligned}$$

Now suppose I do a bunch of different sets of trials, such as practice exams of varying lengths. I want to model each practice exam as being drawn from a distribution of the appropriate size with the same probability. What's the right way to combine them?

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When you compute the Poisson-distributed probability of observing $k$ events in some time interval, you find

$$\begin{aligned} P(k) &= \frac{e^{-λ} λ^k}{k!} \end{aligned}$$

where $λ$ is the mean number of observed events. Here $λ$ needs to be dimensionless, because it's in the argument to the exponent and it's being raise to some power. But if you graduate to the exponential distribution, which describes the waiting time between successive events, then $λ$ acquires units of inverse time:

$$\begin{aligned} \mathrm dP(t) &= λ e^{-λ t} \mathrm dt \end{aligned}$$

Things that magically change units bother me. What's happening here?

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Here's a cute little computation. Suppose you pick two uniform random numbers. What's the probability that their ratio rounds to an even number? It's straightforward enough to get "this is a sum that looks like it's probably famous." But it turns out that different definitions of rounding pull in different transcendental constants.

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