Someone please help me. I couldn't understand the meaning of following question

Please help me. I couldn't understand the meaning of following question and it's solution. This question is from ct3 of iai November diet 2010.

 

First I suppose you need to understand what will be paid by the insurance company and Vivek for each value of K.

• K=0: Insurance Company 0, Vivek 0
• K=1: Insurance Company 0, Vivek 1
• K=2: Insurance Company 1, Vivek 1
• K=3: Insurance Company 2, Vivek 1

Let's call what the insurance company pay L and so E[L] = 0*0.4 + ... + 2*0.1 = 0.40


We're told in the question that:
Profit = Revenue - Insurance Premiums - Other Expenses - Retained Repair Costs

Now, the first three of those are fixed (I'll come on to the repair costs below):
Revenue = 1.70
Insurance Premiums = 110% * E[L] = 110% * 0.40 = 0.44
Other Expenses = 20% * Revenue = 20% * 1.70 = 0.34

So, we now have:
Profit = 1.70 - 0.44 - 0.34 - Retained Repair Costs
Profit = 0.92 - Retained Repair Costs


We need to consider two cases now for the variable part (the retained repair costs):

1. Profit made if Vivek's company do not retain any of the repair costs. Under this option profit = 0.92, and so the dividend = 0.92.
2. Profit made if Vivek's company do retain some repair costs. Under this option profit = 0.92 - 1 = -0.08, so no dividend can be paid, i.e. Dividend = 0.

Notice that (1) can only occur if K=0 (i.e. probability of 0.4), so (2) must occur with probability of 0.6.

Therefore, expected dividend:
0.92*0.4 + 0*0.6 = 0.368


I would hazard a guess that the reason you struggled with this was due to finding the expected value of retained repair costs (= 0.60), then calculating Profit = 0.32.

This doesn't work because you need to work the profits through in each of the possible scenarios to be able to determine what the dividend payment would be. It's really just MAX(Profit, 0), but does add a small complexity.