Alright, let's break this down step by step. We'll be calculating the expected value of this policy for the insurance company. Expected value is like an average outcome, if the scenario was repeated many times. It helps the company to know, on average, how much money they can expect to make (or lose) from selling this policy.
Here are the key numbers we need:
- The insurance company receives $270 for selling the policy.
- They'll have to pay out $280,000 if the person doesn’t survive the year.
- The probability that the 20-year-old female survives the year is 0.999544.
Now, let's look at the two scenarios the insurance company faces:
1. The 20-year-old female survives the year. The probability of this happening is 0.999544. In this case, the company makes $270 (they don't have to pay anything, so they keep the premium).
2. The 20-year-old female doesn’t survive the year. The probability of this is 1 - 0.999544 = 0.000456. In this case, the company loses $280,000 but they got $270, so the net loss is $280,000 - $270 = $279,730.
To calculate the expected value, we'll multiply each outcome by the probability of it happening and sum them up.
Expected value = (Profit if she survives * Probability of surviving) + (Loss if she doesn’t survive * Probability of not surviving)
= ($270 * 0.999544) + (-$279,730 * 0.000456)
= $269.737 + (-$127.45)
= $142.29 (approximately)
This means that, on average, for each policy like this that the insurance company sells, it expects to make around $142.29.
It’s important to understand that this doesn't mean the insurance company will make this amount every time. It's an average over a large number of policies. There will be years where they have to pay out the $280,000, but since the probability of that happening is very low, and they sell policies to many people, they make money in the long run.
In simple terms, it's like the insurance company is betting that the person will survive the year, and the statistics say that they’re likely to win this bet most of the time, so they make money.