Most throws keeping a small-sum chance under half

A pair of fair dice is thrown $n$ times. Let $p$ be the probability that in at least one of these throws the two faces add up to $4$ or less. What is the largest $n$ for which $p<0.5$?

$n=3$. A single throw has sum at most $4$ with probability $\tfrac{6}{36}=\tfrac16$, so $p=1-(5/6)^n$. Since $(5/6)^3\approx0.579>0.5$ but $(5/6)^4\approx0.482<0.5$, the largest valid $n$ is $3$.