Suppose care insurance companies are able to identify some trait that is perfectly correlated with driving skill. So all

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Suppose care insurance companies are able to identify some trait
that is perfectly correlated with driving skill. So all
drivers with the same trait has the same driving skill.
Suppose insurance companies believe that all drivers drive
carelessly.
Suppose they find that half of the drivers with the same
trait as Ben have 3 accidents per year (r = 3) and half of
those drivers have 4 accidents per year (r = 4).
How would insurance companies' belief about Ben's driving
skill depend on Ben's frequency of accidents?
Answer:skilled for sure if r ≤1, skilled with probability 1/2 if
r = 2, and unskilled for sure if r ≥3.
Following question, does Ben have an incentive to drive
normally instead of carelessly?
a.Yes, because driving normally will reduce Ben's
frequency of accidents.
b.No, because by driving normally instead of carelessly, Ben's
payoff will go down by 1, given that the expected value of q
does not change, while cost of effort goes up by only 1.
c.No, because driving normally costs more than driving
carelessly.
d.Yes, because by driving normally instead of carelessly, Ben's
payoff will go up by 2, given that the expected value of q
goes up by 3/8 , while cost of effort goes up by only 1.