 Solution

Let $q$ be the genuine rain probability and $p$ the reported probability. The linear probability score has expected rewardThis is linear in $p$, so its optimal report and expected reward areAt $q=1/2$ every report scores $1/2$ on average. Truthful reporting scores $q^2+(1-q{)}^2$, strictly below the optimum for interior $q\ne 1/2$. Thus the rule is not proper: it rewards maximal confidence in the more likely outcome. The expected total over days is ${\sum }_t\max (q_t,1-q_t)$.