An insurance company classifies its incoming claims as low if they under $10,000$ USD, medium if they are $10,000$ USD or above but under $20,000$ USD, and high otherwise. During the year, $79.2$% of its policyholders filed no claims, $15.9$% filed low claims, $3.4$% filed medium claims and $1.5$% filed high claims.
If a policyholder filed a claim, what is the probability that it was not a low claim?
My attempt:
If it was not a low claim, then it either is a medium claim or a high claim. Thus we want P[either Medium or High] $=$ P[M union H] $=$ P[M] + P[H] $= 0.034 + 0.015 = 0.049$
Is it correct?
We cannot ignore the probability of a policyholder not filing a claim. The question expects you to calculate $\mathbb P(M\cup H|N^c)$, where $M,H,N$ refers to the events where a policyholder files a medium claim, a high claim and no claims respectively, and $c$ denotes the complementary event. We have $$\mathbb P(M\cup H|N^c)=\frac{\mathbb P(M\cup H)}{\mathbb P(N^c)}=\frac{0.034+0.015}{1-0.792}\approx 0.2356$$