There are two groups, group 1 and 2. Group 1 has a 0.5 chance of losing $36, while group 2 has a 0.1 chance of losing $36 dollars. The groups are of equal size. Now an insurance company is willing to cover the losses for a payment. However, the insurance company has an imperfect test for determining what group a person belongs to. The chance that a person actually belongs to the group the insurance thinks they are part of, is X. (0<x<1)
If a person from group 2 is willing to pay a maximum of $4 to the insurance company, what value should X have?
I can't really think of any relevant equations, it seems like a rather straightforward probability question.
The Attempt at a Solution
Assuming that the insurance company had no test to determine who was part of what group, they would assume that the probability of someone losing $36 would be equal to (0.1+0.5)/2 = 0.3; they would ask them to pay $10.8. This is more than the $4 that group 2 members are willing to pay.
However, if the probability is p, then for p = 1/9 members of group 2 will pay the insurance money, as 1/9*36 = $4. However, I don't know how to go to a value of X from here. All I know is that the insurance probably has to think that a large portion of the combined group actually belongs to group 2, to lower the average probability. But whatever I try, I can't seem to come up with a good approach for calculation the value of X. Any push in the right direction would be much appreciated.
Possibly, it could be solved like this:
If the insurance selects someone from group 2, they will ask a price of 0.1X x 36 + 0.5(1-X) x 36. Solving this for $4:
0.1X*36+0.5(1-X)*36 = 4 → 0.1X+0.5-0.5X = 1/9 → 0.4X = 0.5 - 1/9
So for X = 0.972, group 2 will pay the insurance money
Is that correct?