# Economics Practice Homework Problem

Working on probability theory in an economics class. This was on of the practice problems that I'm struggling to back up with math. Any help would be appreciated! :)

• An accident insurance company has reserves to cover claims on no more than 3 accidents. If all accidents have a probability of 0.01 of occurring, if all accidents are independent, and if the insurance company writes 100 policies, what is the probability that they will not have enough reserves?
• @Andy: Have you tried anything so far? If so, it might be helpful to include it in your post. – Bey Nov 10 '10 at 5:56
• Hint: Poisson distribution. – Yuval Filmus Nov 10 '10 at 6:05

Let's $X_i$ be a Bernoulli variable that is 1 if an accident occured with the policy number $i$, 0 otherwise. Thus we have: $$P(X_i = 1) = 0.01, P(X_i = 0) = 0.99$$ This is true for all $i$ since we assume independance.

Let $Y = X_1 + X_2 + \dots + X_{100}$ the sum of all $X_i$. $Y$ counts the number of accidents that occured.

That means you are looking for $P(Y > 3)$.

Now, if you are having probability class, you should have seen what a Bernoulli variable is, and what the distribution of a variable that is the sum of independant Bernoulli variables is.

If you don't know, look here.

Please note that according to your question I had to made the assumption that the probability you give (0.01), is the probability that for 1 policy there will be 1 accident and that there cannot be more than 1 accident for the same policy. Usually, we give a probability that in a given amount of time there will be an accident.