I’m a little stumped by this question:
A fair coin is flipped repeatedly until we get “H” exactly 120 times. We’ll mark with $X$ the number of flips in which we got the result “T”. approximate with CLT $P(X \geq 137)$
my progress so far:
had I known the total number of coin flips I would’ve been able to answer the question with relative ease. I’ve started as I usually do by defining $X_i$.
$$X_i = \begin{cases}1 & \text{The ith flip landed on “T”} \\ 0 & \text{Otherwise}
\end{cases} $$
I know $$X = \sum_{i=1}^{n=X+120} X_i$$ and finding $\mu$ and $\textsf{Var}(X_i)$ is doable as well: $E(X_i) = P(X_i=1) = 0.5$ and $\sigma = \sqrt{p \cdot q} = \sqrt{0.5^2} = \frac12$.
I’m starting to experience problems trying to get to the form of CLT - I can’t rid the right expression of X.
$$P\left(\frac{X - \mu n}{\sqrt{n}\sigma} \geq \frac{137 -\frac12(X+120)}{\frac12 \sqrt{X+120}}\right)$$
I don’t know how to continue from here.