Upper bound lower tail of binomial tail

Question

I have a random variable $Y\sim Bin(n,p)$ and I know that $E[Y]<20$. Now I want to compute $P[Y\leq60]$.

I try to compute it by firstly compute $P[Y > 60]$ using Chernoff bound, i.e. $P[Y\geq (1+\delta)\mu)\leq e^{-\frac{\delta^2 \mu}{3}}$.

However, I can't find how I can use that $E[Y]<20$ here. How can I approach this?

Answer 1

A simpler approach is to note that if $np < 20$, $\Pr[Y > 60]$ is maximized when $n$ is as large as possible. In the limit where $\lambda = np = 20$ and $n \to \infty$, we have $Y \sim \operatorname{Poisson}(\lambda = 20)$ and $$\Pr[Y > 60] < 1 - \sum_{y=0}^{60} e^{-\lambda} \frac{\lambda^y}{y!} \approx 1.37557 \times 10^{-13}.$$