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?
  • $\begingroup$ @Andy: Have you tried anything so far? If so, it might be helpful to include it in your post. $\endgroup$ – Bey Nov 10 '10 at 5:56
  • 1
    $\begingroup$ Hint: Poisson distribution. $\endgroup$ – 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.


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