<s>[INST] Asymptotics of probability of Newton-Pepys problem In Newton-Pepys problem one is interested in probability $p_n$ of getting at least $n$ sixes in $6 n$ independent throws of regular 6-sided die. 
The number of sixes $S_m$ obtained in $m$ throws follows a Binomial distribution  $\operatorname{Bin}(m,p)$, where $p=\frac{1}{6}$ is the probability of getting a six in a single throw. Thus
$$
     p_n = \Pr\left(S_{6n} \geqslant n\right)
$$
Notice that $\mathsf{E}(S_{6n}) = 6 n p = n$, and variance $\mathbb{Var}(S_{6n}) = \sqrt{5 n/6}$, hence in the large $n$ limit
$$
    \lim_{n \to \infty} p_n = \lim_{n \to \infty} \Pr\left(\frac{S_{6n}-n}{\sqrt{5 n/6} }\geqslant 0\right) \stackrel{\text{CLT}}{=} \Pr(Z \geqslant 0) = \frac{1}{2}
$$
In fact $p_n$ is monotonically decreasing sequence:

Q.: How can one find the large $n$ asymptotics of $p_n$?
 [/INST] The probability $p_n$ has a representation in terms of regularized beta-function, which incidentally is the cumulative distribution function of beta distribution.
$$
   p_n = \Pr(S_{6n} \geqslant n) = I_{p}\left(n, 5n+1\right) = \Pr(X < p ) = \Pr\left(X < \frac{1}{6} \right))
$$
where $X$ follows $\mathcal{Be}\left(n,5n+1\right)$. It is well-known that $X \stackrel{d} = \frac{Y}{Y+Z}$ where $Y$ and $Z$ are independent random variables following gamma-distirbutions $\Gamma(n,1)$ and $\Gamma(5n+1,1)$ respectively.
$$ \begin{eqnarray}
  p_n &=& \Pr\left(\frac{Y}{Y+Z} < \frac{1}{6}\right) = \Pr(5 Y < Z) = \Pr\left(5 (Y-n) < Z -5 n\right) \\ &=& \Pr\left(5 \frac{Y-n}{\sqrt{n}} < \frac{Z -5 n}{\sqrt{n}}\right) = \Pr\left(5 \frac{Y-n}{\sqrt{n}} < \frac{Z -5 n-1}{\sqrt{n}} + \frac{1}{\sqrt{n}}\right) \\
  &=& \Pr\left(5 \frac{Y-n}{\sqrt{n}} < \frac{Z -5 n-1}{\sqrt{5n+1}} \sqrt{5 + \frac{1}{n}} + \frac{1}{\sqrt{n}}\right)
\end{eqnarray}
$$
Since mean and standard deviation of $Y$ equal $n$ and $\sqrt{n}$ and mean and standard deviation of $Z$ equal $5n+1$ and $\sqrt{5n+1}$.
Approximating $\frac{Y-n}{\sqrt{n}}$ and $\frac{Z -5 n-1}{\sqrt{5n+1}}$ with iid standard normal random variables:
$$
   p_n \approx \Pr\left(5 Z_1 - Z_2 \sqrt{5+\frac{1}{n}} < \frac{1}{\sqrt{n}}\right) = \Phi\left(\frac{1}{\sqrt{30 n + 1}}\right)
$$
Unfortunately, the above approximation is not very accurate.
</s>