Probability: What is the best solution for this problem about Discrete distributions?

Question

Here is the problem:

The drainage system of a city has been designed for a rainfall intensity that will be exceeded on an average once in 50 years. What is the probability that the city will be flooded at most 2 out of 10 years?

May you tell me what do you think about my solution?

In my perspective, we can use the Binomial or the Poisson Distribution.

However, I think that the precise one, is the Poisson because of this explanation of my book, the ninth edition of Probability and Statistical Inference written by Robert Hogg, Elliot Tanis and Dale Zimmerman.

Please read it:

In despite of that, the same book talks also about the Binomial Distribution.

Please read it:

Which solution would be the best solution?

Thank you so much for your help!

Answer 1

You have chosen two promising possible models. I botched my first try at a good explanation, but was kindly nudged onto the right track in Comments by @lulu.

Poisson. If the rate of floods is $1$ per 50 years, then it is $\lambda_{10} = 1/5 = 0.2$ per ten years. Thus, let $X \sim \mathsf{Pois}(\lambda_{10} = 1/5),$ which has $E(X) = 1/5.$ Then using the formula for the Poisson PDF (or PMF) one gets $P(X \le 2) = 0.9989.$ or essentially a 'sure thing'. The computation in R statistical software is shown below:

ppois(2, .2)
## 0.9988515

Binomial. Suppose we try modeling with a binomial random variable $Y.$ Because a 10-year time span is envisioned, let $n = 10.$ We need $E(Y) = np = 10p = 1/5,$ so $p = 1/50 = 0.02.$ Then $P(Y \le 2) = 0.9991 \approx 1,$ which is nearly the same result as for the Poisson model.

pbinom(2, 10,.02)
## 0.9991361

I used R statistical software to do the computations, but you should probably use the formulas for Poisson and binomial probabilities.

Here is a plot with the Poisson model represented as bars and the binomial model represented as red circles. For both models, almost all of the probability rests on 0 or 1 serious floods. (I had to make a tall plot for the probability of two storms in 10 years to show on the graph.)

Note: Binomial distributions can often be approximated by Poisson distributions with the same mean. In the case where $\mu = np$ remains constant, while $n \rightarrow \infty$ (and $p$ shrinks appropriately) it can be shown that the PDF of $\mathsf{Binom}(n,p)$ converges to that of $\mathsf{Pois}(\mu).$ [An essential element of the proof is that $(1 - a/n)^n \rightarrow e^{-a}.]$