For every $m$ and $k$ integers with $k$ odd, denote by $\left[\frac{m}{k}\right]$ the integer closest to $\frac{m}{k}$. For every odd integer $k$, let $P(k)$ be the probability that

$\left[\frac{n}{k}\right]+\left[\frac{120-n}{k}\right]=\left[\frac{120}{k}\right]$

for an integer $n$ randomly chosen from the interval $1\le n\le 119!$. What is the minimum possible value of $P(k)$ over the odd integers $k$ in the interval $1\le k\le 119$?