When a war starts in a region, civilians usually flee to the borders. In an evac

Question

When a war starts in a region, civilians usually flee to the borders. In an evacuation center, an average of 1800 civilians arrives per day (they only arrive during day time, roughly between 7 AM - 7PM). Additional supplies are brought to the evacuation centers every day. a) What is the average number that should be processed in the registration area per hour? Processing includes registration of the person, interview, and medical checkup. b) Certain additional supplies could spoil if not used within 24 hours. But it is also important that no one will be left without supplies so the workers allocate 1850 sets per day. What is the probability that it will be insufficient, that is, at least one person would be left without supplies? c) Recommend a quantity of supplies such that the probability of stock-out is at most one percent.

Explanation / Answer

Solution

Let X = number of arrivals per day (7AM to 7 PM i.e., 12 hours). Then, X ~ Poisson (), where = average number of arrivals per day, given to be 1800. So, = 1800 per day.

Back-up Theory

Poisson probabilities can be approximated by Normal Probability by the rule,

P(X > < t) = P[Z > < {(t - )/ }], where Z ~ N(0, 1).

Part (a)

Since average arrivals are 1800 per day and the day is given to extend from 7 AM to 7 PM, average arrivals per hour = 1800/12 = 150. So, average number that should be processed in registration area should be 150 per hour ANSWER

Part (b)

Since allocation of supplies per day is given to be 1850 sets, at least one person would be left out without supplies if the number of arrivals is greater than 1851. So,

P(at least one person would be left out without supplies) = P(X > 1851).

Now, vide Back-up Theory, P(X > 1851) = P[Z > {(1851- 1800)/1850}] = P(Z > (51/43.0116)

= P(Z > 1.1857) = 0.11781 [using Excel Function ]

Thus, P(supplies is insufficient) = 0.11781 ANSWER

[Note: just for seeing how close the approximation is, the exact Poisson probability using Excel Function is 0.1127]

Part (c)

Let t be the supplies that would ensure that stock-out is at most 1%. Then, we should have:

P(X > t) 0.01.

Again, vide Back-up Theory, P(X > t) 0.01 => P[Z > {(t - 1800)/1850}] 0.01

=> (t - 1800)/43.0116 = 2.326348 [using Excel Function] or t = 1900.

So, quantity of supplies so that probability of stock-out does not exceed 1% is:

1900 sets ANSWER

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