Ben just bought a car and signed a car insurance contract. In a year, this contract will expire and Ben will move across

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Ben just bought a car and signed a car insurance contract. In a
year, this contract will expire and Ben will move across the
country. Suppose, when he is in the market for a new car
insurance contract next year after he moves, the potential
insurance companies can all see the number r of car accidents
Ben has been involved in this year. This number r depends on
Ben's driving skill θ , Ben's effort e, and the road condition ε.
More specifically, r = ε - θ - e. Ben's driving skill θ is
unknown to both Ben and all potential car insurance companies.
All believe that Ben's driving skill θ is 0 (unskilled) with
probability 1/2 and 1 (skilled) with probability 1/2 .
The road condition ε is 3 with probability 1/2 and 4 with
probability 1/2 . Ben can choose to drive very carefully (e = 2),
normally (e = 1) or carelessly (e = 0). But none of the
potential car insurance companies observe Ben's effort choice.
The potential car insurance companies will offer the car insurance
contract that best matches Ben's driving skill. The more
likely they think Ben is skilled, the lower the insurance
premium they would charge. Ben wants to minimize the
insurance premium, but driving carefully is costly. In
particular, given the potential insurance companies' belief
that Ben is skilled with probability q, Ben's payoff is equal to 8q
- c(e), where c(e) = 0 if e = 0, c(e) = 1 if e = 1 and c(e) =
4.5 if e = 2. Use the above information to answer all of the
following question.
Suppose car insurance companies believe that Ben drives
carelessly. How does their belief about the probability Ben is
a skilled driver (θ = 1) depend on Ben's number of accidents
r?
a.skilled for sure if r = 0, skilled with probability 3/4 if r =
1, skilled with probability 1/2 if r = 2, skilled with
probability 1/4 if r = 3, and unskilled for sure if r ≥ 4.
b.skilled for sure if r ≤ 2, and unskilled for sure if r ≥
3.
c.skilled for sure if r ≤ 2, skilled with probability 1/2 if r =
3, and unskilled for sure if r =4.
d.skilled for sure if r ≤ 1, skilled with probability 1/2 if r =
2, and unskilled for sure if r ≥ 3.