You are a homeowner and your township has sent you a notice that you must fix your sidewalk. You will pay a penalty of 100 per day for each day until the sidewalk is fixed, not including today and the day the sidewalk is fixed.
You have called a sidewalk repair company to fix the sidewalk. The company will not come today. The probability that the company fixes the sidewalk on any day starting tomorrow, given that it has not already been fixed, is 0.8. The probability of fixing the sidewalk on any day is independent of the probability of fixing the sidewalk on any other day.
You are given
Calculate the expected amount of penalties you pay.
Here is my answer and reasoning:
The homeowner pays penalties if the company does not show up.
Thus the expected value is
100(0.2)+ 100(0.2)^2 + 100(0.2)^3 + 100(0.2)^4 + .... =
Thus my answer is 25.
However, I did not use the formula that was given. How can we use that formula to solve the problem? The author is trying to teach the Negative Binomial. However, I do not think that I used that, did I? Thank you for your help!
To get my answer I used this formula:
Edited for details: The probability you are charged \$100 on a particular day will be if the company did not show up that day. So, if the company does not show up on day one, but does show up on day 2, you are charged just the \$100 for the day the did not show as per your assumptions. The probability of this event is $(0.2)(0.8)$. Similarly if they missed the first two days and show on the third, you are looking at \$200 with a probability of $(0.2)(0.2)(0.8)$.
The fee will be 100 with probability $0.2(0.8)$.
The fee will be 200 with probability $0.2^2(0.8)$.
The fee will be 300 with probability $0.2^3(0.8)$.
etc.
Therefore the expected value will be
$$\sum_{i=1}^{\infty} 100n\cdot 0.2^n(0.8) = 80\sum_{n=1}^{\infty} n\cdot 0.2^n$$
Hence the given formula.