Estimate the probability such that if 2 coins are thrown at the same time, for 100 times, "heads" to appear a number of times in between $15$ and $20$.
I considered $X_k= \begin{pmatrix}0&1\\0.5&0.5\end{pmatrix}$ so $M[X_k]=0.5$ and $\sigma=0.25$, rather than considering $2$ same events at the same time i thought it's the same to consider the same event twice the number of throwns so $n = 200$.
Then $$P(15 \leq X_k\leq 20)= \Phi(\frac {20 - 200\times0.5}{\sqrt{200\times0.5\times 0.5}}) - \Phi(\frac {15 - 200\times0.5}{\sqrt{200\times0.5\times 0.5}})=\Phi(12.14)-\Phi(11.42)$$
But these values are not in the table and the tables goes up to $3.5$ how to make them be in the table? How do i proceed?
If you want a very accurate value, you can use a calculator's $\verb!normalcdf!$ function, or find something similar online. However, note that $\Phi(12.14) \approx 1$ and $\Phi(11.42) \approx 1$, meaning that the probability that $X_{k}$ lies between $15$ and $20$ will be about zero.
This makes sense intuitively because it's very unlikely to only get between $15$ and $20$ heads if you're flipping $200$ coins.