How many beds should the hospital have so that the probability that a student will be turned away for lack of bed is less than $0.01 $

Question

At a certain men's college, the probability that a student selected at random on a given day will require a hospital bed is $\dfrac {1}{5000}.$ If there are $8000$ students, how many beds should the hospital have so that the probability that a student will be turned away for lack of bed is less than $0.01 $

Attempt: Let $X$ be the random variable which denotes the number of students requiring a hospital bed on a given day.

Trying to use the poisson distribution, we obtain, $\lambda = 8000 \cdot \dfrac {1}{5000} = \dfrac {8}{5}$. Then, using poisson distribution : $P(X=x) = \dfrac {e^{-\lambda}\cdot \lambda^x}{x!}$

Let $n$ be the required number of beds ( $n < 8000)$. Then, by the given requirement, the probability that $n$ beds will be occupied should be less than $0.01 \implies \dfrac {e^{-\frac{8}{5}}\cdot \big( \frac{8}{5}\big)^n}{n!} \le 0.01$

$\implies \Big ( \dfrac{8}{5} \Big)^n \cdot \dfrac{1}{n!} \le 0.01 \cdot e^{\frac{8}{5}}$

Solving for which we get $n \ge 6$.

Is this the correct approach? Thanks a lot for your help.

Answer 1

$X$ is binomial and we try to find the minimum $n$ that satisfy the following: $P(X>n)\leq 0.01$, i.e. number of ill students is larger than number of beds in the hospital. This is equivalent to $P(X\leq n)\geq 0.99$. $$P(X\leq n)=\sum_{i=0}^n P(X=i)=\sum_{i=0}^n {8000\choose i}\left(1\over 5000\right)^i\left(4999\over 5000\right)^{8000-i}$$ This is numerically challenging (still can be done using softwares like Matlab). A better way is using Poisson approximation since $n$ is large and $p$ is small. The rate parameter $\lambda$ is defined as $np=8/5$. Substituting into the summation above: $$P(X\leq n)=\sum_{i=0}^n P(X=i)=\sum_{i=0}^n \frac{e^{-\lambda}\lambda^i}{i!}=e^{-\lambda}\left(1+\lambda+\frac{\lambda^2}{2}+\frac{\lambda^3}{6}...+\frac{\lambda^n}{n!}\right)$$

A couple trials will reveal that you actually need $n=5$ beds at minimum. Your attempt lacks the CDF approach.

Answer 2

Comment continued: In case the 'search' implied by the brackets [ ] in R is not clear, and in order to show how well the Poisson approximation to the binomial distribution works in this example, here is a table of relevant Poisson and binomial CDFs.

x = 1:10; p.pr = ppois(x, 8/5); b.pr = pbinom(x,8000,1/5000)
cbind(x, p.pr, b.pr)
      x      p.pr      b.pr
 [1,]  1 0.5249309 0.5249116
 [2,]  2 0.7833585 0.7833688
 [3,]  3 0.9211865 0.9212058
 [4,]  4 0.9763177 0.9763310
 [5,]  5 0.9939597 0.9939657  # first entry exceeding 0.99
 [6,]  6 0.9986642 0.9986663
 [7,]  7 0.9997396 0.9997401
 [8,]  8 0.9999546 0.9999548
 [9,]  9 0.9999929 0.9999929
[10,] 10 0.9999990 0.9999990