# Central Limit Theorem application to discrete random variable

A fair coin is tossed until 100 heads appear. Find the probability that at least 226 tosses will be necessary.

Solution: Let $X$ be the geometric random variable with $p = ½$. Then $S_{100} = X_1 + ... + X_{100}$ is a random variable of the number of coin tosses that result in 100 heads, and $ES_{100} = 100 *(1/p) = 200, Var S_{100} = 100(1-p)/p^2 = 200$.

\begin{align} P(S_{100} \geq 226) &= 1 - P(S_{100} \leq 225) \\[8pt] &\approx 1 - \Phi\left( \frac{225 - 200}{10\sqrt{2}}\right) \\ &= 1 - 0.96145\\ &\approx 0.03855 \end{align}

The book's solution is 0.0415. I'm not sure what I'm doing incorrectly.

• Looks ok to me (though I would use 225.5 - which gives me 0.3568). Are you sure you got all right? (and the book is reliable?) – leonbloy Aug 4 '14 at 18:00
• @leonbloy The book is reliable. I have not seen any error in the solution. Is there a different way to solve this problem, so that I can cross check? – user90593 Aug 4 '14 at 18:02
• Yes, see here: sosmath.com/CBB/viewtopic.php?t=33427 – leonbloy Aug 4 '14 at 18:05