Approximation for probability of at least $t$ events

Question

I'm reading through a paper, and they are discussing the approximate probability that $t+1$ out of $t^b$ events occur, where $b$ is a constant, and the probability of each event occurring is independent, and of value $\epsilon$. The author says this value is approximately $(t^b\epsilon)^{t+1}$, but I fail to see why this is the case; any Taylor expansion I try to use just seems too messy. We can assume that $\epsilon$ is small, does anyone have any suggestions?

Answer 1

Perhaps the author is thinking in terms of a Poisson approximation. If $X$ is the number of events among the $t^b$ possibilities that occur, then $X$ is binomially distributed with parameters $(t^b,\epsilon)$.

When $t^b$ is large, and $\epsilon$ is small, the Poisson distribution with parameter $\lambda=t^b\epsilon$ approximates the binomial distribution, so if $Y$ is a random variable having this distribution, we get

$$P(Y=t+1) = e^{-t^b\epsilon}\frac{(t^b\epsilon)^{t+1}}{(t+1)!}.$$

Now you just need to throw away whatever doesn't fit! Kidding aside, if $t^b\epsilon$ is small, the exponential factor is close to $1$, which may justify getting rid of it. Also, it is possible that $(t+1)!$ is not a big deal compared to the numerator, again depending on the size of the quantities involved. This is one possibility of what he's been thinking, but when there's no more information given, I urge you to check how the true distribution matches up to the approximation in the paper in some examples that are reasonable in the given setup.

You can also consider if the form of the approximation is important for the rest of the paper, or if, in your mind, you can think of the true distribution instead, or a better approximation than the one given.