Proof:
1. Suppose $\sqrt{2}=\frac{p}{q}$
Square both sides: (50% chance of success)
2. Then 50% of the time, $2=\frac{p^2}{q^2}$
Rearrange (40% success rate)
3. Thus 50%*40% of the time $2q^2=p^2$
even*even=even, even*odd=even, odd*odd=odd (100% success rate)
4. Thus $p^2$ is even, thus p is even, thus $q^2$ is even thus $q$ is even, contradict the fact that $p,q$ have no common factors
Thus 20% of the time, there is a contradiction