Final answer:
The expected value to the insurance company of a single policy for ramp models, who have a 99.97% chance of being uninjured in one year, is P99820. This is calculated by multiplying the premium by the probability the model is uninjured and subtracting the product of the pay out and the probability the model is injured.
Step-by-step explanation:
To calculate the expected value for the insurance company of a single policy for ramp models, we need to consider two scenarios: the model being uninjured and the model getting injured and the policy having to pay out.
Firstly, the probability of a model being uninjured in one year is 99.97%, and the insurance company will keep the full premium in this case. Secondly, the probability of a model being injured and the policy having to pay out is 0.03% (which is 1 - 99.97%).
Let's denote the premium paid by the models as P and the pay out for the injury as L.
- Probability of being uninjured: 99.97%, which equals 0.9997.
- Probability of being injured: 0.03%, which equals 0.0003.
- Expected value from a model being uninjured: P * Probability(being uninjured) = P * 0.9997.
- Expected value of having to pay out for an injury: -L * Probability(being injured) = -L * 0.0003.
The total expected value for the insurance company is the sum of the expected values for each scenario:
Expected value = P * 0.9997 - L * 0.0003.
Substituting P = P¹⁰⁰⁰⁰⁰ and L = P⁵⁰⁰⁰⁰⁰.⁰⁰, we get:
Expected value = P¹⁰⁰⁰⁰⁰ * 0.9997 - P⁵⁰⁰⁰⁰⁰ * 0.0003.
By doing the calculations:
Expected value = 100000 * 0.9997 - 500000 * 0.0003.
Expected value = 99970 - 150.
Expected value = 99820.
So, the expected value to the company of a single policy is P99820.