Insurance doesn't give full payout

The value x, of an **individual’s wealth**, excluding the value of his car is **£35,000**.

His utility function is

**U(x) = ln (x + 300)**

There is a **2%** probability of theft or accident reducing the value of his car, worth **£15,000, to a scrap** value of zero.

An insurance company is considering insurance for the car up to a **maximum payout of £5,000** at a **premium of £200**. Show that this insurance arrangement would be acceptable both to the company and the individual.

**The company**'s utility function is,

**U(x) = 2ln(x)**,

where x is the company’s **assets, £2,000,000**.

I can work out that the insurance company would accept the deal, however I don't know how to work it out for the individual.

P: E[U(a-x)] = U(x - P)

as E[U(a-x)] is the part without insurance i assume i have to do something with the U(x - P) part

If it repayed the full amount i would just do

ln(50,000+300 - P)

Do i have to incorporate the fact that 98% of the time i won't use insurance, however if the insurance is used 2% of the time there is still a loss of 10,000?

my logical step is to do

E[U(a-x)]=10.81867791=(ln(50000+300 - P) * 0.98) + (ln(50000+300-10000 - P)

However this works P out to be 154.98846, which is less than 200, so must be wrong as the question states that the individual accepts the 200 offer.

Please help :)