An insurance company is selling a policy for motorcyclists. It will cover up to $20000$ dollars if the motorcyclist dies within two years. This policy will cost the policy holder $300$ dollars and the probability of death within those two years is $0.035$. What is the expected gain for this company?
So the way I approached this question is to do:
$((300) * (1 - 0.035)) - ((20000) * (0.035))$ = $-410$
My logic being, they will gain based off the $300$ dollar claim multiplied by the probability the motorcyclist survives, and they will lose money based on the $20000$ claimed when the motorcyclist dies. However, the answer to this specific question is $289.5$ dollars. Can anyone explain how this number is arrived at?
Let $G$ be a random variable denoting the gain/profit of the company (in dollars). We want to compute $E[G]$, the expected gain of the company. Let $V$ denote the possible values $G$ can take on; there are two cases to consider:
1) No deaths: $G = 300$, because the policy holder pays for the policy and never dies. This happens with probability $1 - 0.035 = 0.965$.
2) Death: $G = 300 - 20000 = 17300$, since the policy holder pays for the policy BUT the company needs to reimburse with $20000$ dollars. This happens with probability $0.035$.
So, $G$ takes on a value in the set $V = \{300, -19700\}$. The expected gain can be written as so:
$E[G] = \sum_{g \in V} g\cdot P(G = g) = 300 \cdot P(G = 300) -19700 \cdot P(G =-19700)$
Plugging in the approriate numerical values for the probabilities:
$=300\cdot0.965 -19700 \cdot 0.035 = -400$
I think the discrepancy in your answer lies in assuming the company's gain upon a death is $-20000$. You need to account for the fact that the policy holder had already paid $300$, so the company starts at a $+300$ dollar value and pays $20000$, leaving it at $-19700$.