Imagine tossing a coin 50 times. What are the probabilities of observing heads 25 times (i.e., 25 successful experiments)?
The trial of interest consists of 50 separate experiments; therefore, $n=50$. Considering the case of 25 successful experiments where $j=25$. The probability $\left(P_{25}\right)$ is
$$
\begin{aligned}
P_{25} & =C(n, j)\left(P_E\right)^j\left(1-P_E\right)^{n-j} \\
& =C(50,25)\left(P_E\right)^{25}\left(1-P_E\right)^{25} \\
& =\left(\frac{50 !}{(25 !)(25 !)}\right)\left(\frac{1}{2}\right)^{25}\left(\frac{1}{2}\right)^{25}=\left(1.26 \times 10^{14}\right)\left(8.88 \times 10^{-16}\right)=0.11
\end{aligned}
$$