HMMT-Nov 2010 thm, Problem 8

Allison has a coin which comes up heads $\frac{2}{3}$ of the time. She flips it 5 times. What is the probability that she sees more heads than tails?

Solution: The probability of flipping more heads than tails is the probability of flipping 3 heads, 4 heads, or 5 heads. Since 5 flips will give $n$ heads with probability $\left(\begin{array}{l}5 \\ n\end{array}\right)\left(\frac{2}{3}\right)^{n}\left(\frac{1}{3}\right)^{5-n}$, our answer is $\left(\begin{array}{l}5 \\ 3\end{array}\right)\left(\frac{2}{3}\right)^{3}\left(\frac{1}{3}\right)^{2}+\left(\begin{array}{l}5 \\ 4\end{array}\right)\left(\frac{2}{3}\right)^{4}\left(\frac{1}{3}\right)^{1}+\left(\begin{array}{c}5 \\ 5\end{array}\right)\left(\frac{2}{3}\right)^{5}\left(\frac{1}{3}\right)^{0}=\frac{64}{81}$. $\fbox{\frac{64}{81}}$.

Answer: \frac{64}{81}