[Combinatorics]
Problem: A fair coin is flipped $n$ times. Compute the smallest positive integer $n$ for which the probability that the coin has the same result every time is less than $10 \%$.

Solution: After the first throw, the probability that the succeeding $n-1$ throws have the same result is $\frac{1}{2^{n-1}}$. Thus $\frac{1}{2^{n-1}}<\frac{1}{10} \Rightarrow 2^{n-1}>10 \Rightarrow n-1 \geq 4$, so $n=5$ is the smallest possible value.

Final answer: 5