$\text{Description of the following question:}$
Suppose $X$ is the number of successes out of $n$ trials, where the trails are independent of each other. The probability of success at every trial is $p$. The probability that there will be exactly $k$ successes out of $n$ trials is
$$\mathbb{P}(X=k)=\begin{pmatrix}n\\k \end{pmatrix}p^k(1-p)^{n-k},k=0,1,\dots,n$$
The expected number of successes is $\mathbb{E}(X)=np$. If $n\to \infty$ and $p\to 0$
$$\begin{pmatrix}n\\k \end{pmatrix}p^k(1-p)^{n-k}\to e^{-\lambda}\frac{\lambda^k}{k!},k=0,1,\dots,$$ where $\lambda=np$.
Each time a gambler plays a game, there is a one-in-million chance of winning. The gambler plays the game one million times. Find the probability of winning the game zero times, i.e. find the probability of the event that the gambler will lose all one million times that she/he will try.
Note: $1$ million =$10^6$.
