I have this problem I am doing and I cannot solve it. If anyone could help please and thank you!
Background
Police dispatchers call centres staffed to respond to calls from the public. In one particular call centre with 100 dispatchers answering calls, it was observed on average the proportion of individuals absent from work on any particular day was $3\%$. However, on-call staff make up for any absences. There were 4 additional dispatchers available on call everyday at that can fill in for the absences at any notice.
The on-call staff are paid a rate of $350 per day regardless whether or not they are called. Therefore, dispatchers not called in for work are designated as "unecessary spending".
Derive the expected "unecessary cost" per day.
My approach
I am really having trouble constructing this problem. I am unsure of my approach conceptually makes sense.
I have let $\alpha$ be the number of dispatchers not working on any given day and let $\beta$ be the number of dispatchers not needed (i.e. "unecessary") on that same day. I understand I must somehow relate connect the values of $\alpha$ and $\beta$ and determine $E(\beta)$
Any help here would be appreciated.
Updated
To see if my approach works, I have computed the probability there are 3 or less people not working as 4 or more individuals not working suggests all on-call staff will be not "unecessary".
$$ Pr(X\le3) = Pr(X = 3) + Pr (X = 2) Pr(X = 1) + Pr(X=0)\\ = 0.6427 $$
Then, computing the expected value of $\mathbb{E}[Y]$
$$ \mathbb{E}[Y] = n \cdot p \\ = 4\cdot 0.6427 \\ \approx 2.58 $$
Computing the cost now by multiplying 350 gives $\$903$ per day. Is this approach the correct approach?
Thank you again!
No, your approach is not quite correct.
Since you want the expected daily "unnecessary costs" and this is equal to $350$ times the number of on-call staff who are not called in to replace any absent dispatchers, the idea is to define a random variable, say $X$, that counts the number of on-call staff who are not called in. Then $$X \in \{0, 1, 2, 3, 4\},$$ where for instance if $X = 0$, this means every on-call staff member is called in, and there were at least $4$ absent dispatchers that day. Conversely, if $X = 4$, then every dispatcher showed up.
This leads us to define another random variable, say $Y$, that counts the number of absent dispatchers. So we have
$$X = \begin{cases}4 - Y, & Y \in \{0, 1, 2, 3\}, \\ 0, & Y \in \{4, 5, \ldots, 100\}. \end{cases}$$
Then if we make the assumption that each dispatcher's probability of not showing up that day is $p = 0.03$ and is independent of whether any other dispatchers are absent, we can model $Y$ with a binomial distribution with parameters $n = 100$ and $p = 0.03$. Hence the expectation of the number of unused on-call staff is
$$\begin{align} \operatorname{E}[X] &= \sum_{x=0}^4 x \Pr[X = x] \\ &= \Pr[X=1] + 2\Pr[X=2] + 3\Pr[X=3] + 4\Pr[X=4] \\ &= \Pr[Y=3] + 2\Pr[Y=2] + 3\Pr[Y=1] + 4\Pr[Y=0] \\ &= \binom{100}{3}(0.03)^3(0.97)^{97} + 2 \binom{100}{2}(0.03)^2(0.97)^{98} + 3 \binom{100}{1}(0.03)(0.97)^{99} + 4(0.97)^{100} \\ &= (0.97)^{97} \left(161700(0.03)^3 + 9900(0.03)^2(0.97) + 300(0.03)(0.97)^2 + 4(0.97)^3\right) \\ &\approx 1.3092. \end{align}$$
This means the expected "unnecessary cost" per day is $$350\operatorname{E}[X] \approx 458.22.$$
That said, this is not a particularly realistic problem, since the cost is not really "unnecessary." It is the cost of ensuring that there is at least an $\Pr[Y \le 4] \approx 81.7855\%$ probability that there will be enough on-call staff to cover all absences on a given day. A more realistic modification to this problem would incorporate another cost for not having enough on-call staff; e.g., if there is at least one uncovered absence, the company is fined $10000$ that day. Then we might ask how many staff should be kept on-call to minimize the expected daily loss. Should the company have only $4$? Or is a smaller or larger number more optimal?