We throw a fair coin until we have 50 heads. What is the approximate probability that we flip more than 120 times?
I guess I can rewrite it so as to ask what's the probability that in 120 flips I have less than 50 heads. So, we have iid random variables $X_i =\begin{cases}1 \ \ \ \text{if heads} \\ 0 \ \ \ \text{if tails}\end{cases}$
$\Bbb P(\frac{\sum\limits_{i=1}^{120}X_i-\Bbb nE[X_i]}{\sqrt{nVar(X_i)}}<\frac{50-\Bbb nE[X_i]}{\sqrt{nVar(X_i)}})=\Bbb P(Z<\frac{50-\Bbb 120\cdot 1/2}{\sqrt{120\cdot 1/4}})=\Bbb P(Z<-1.826)\approx0.0339.$
Am I correct?