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Running a Blood Bank

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You are running a blood bank. Every month, a random number of hospital patients will need blood that you will supply. Also, every month, a random number of donors will come in and give blood. You goal is to provide enough blood on hand to supply those patients who need it.

At the beginning of the current month you have no pints on hand. Your forecast for this month predicts that the number of pints needed by patients, denoted by N for needed, is normally distributed with mean 35 and standard deviation 8.

Your forecast also predicts that the number of pints given by donors who will come in this month to donate blood, denoted by D for donated, is a normally distributed with mean 20 and standard deviation
15. The amounts of pints donated and required are assumed to be independent from one another.

If the amount of blood needed exceeds the amount of donated blood, the amount of blood you 'have' at the end of the month is negative representing the deficit. (For example if this month 25 pints donated and 30 pints needed, the amount of blood you have in your bank at the end of the month is 25 - 30 = -5 pints. In general, the amount of blood you have in you bank at the end of the month is D-N)

(a) (3 points) What is the likelihood that the amount of the blood donated this month exceeds 45, i.e., Pr(D>45)?

(b) (3 points) What is the probability distribution describing the amount of blood you have in your bank at the end of the month? Compute its mean and variance.

(c) (3 points) What is the probability that you'll run out of blood by the end of the month, i.e., Pr(D-N<0)?

(d) (4 points) If you don't run out of blood at the end of the month, you pay no additional cost. If you do run out of blood before the end of the month, you will pay a fixed cost of \$1000 for an emergency shipment of blood (paying this flat fee will allow you to have all the pints of blood that you need).

You have the option of spending \$100 on a marketing campaign, which will definitely bring in all the donors you need (but you don't want to spend this money if you don't need it).

Draw a decision tree describing the situation you're facing, calculate the EMV, and make a recommendation.

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Solution Preview

mu_N = 35
sigma_N = 8

mu_D = 20
sigma_D = 15

(a)
P(D>45) = ?
x = 45
z = (x - mu_D)/sigma_D = (45 - 20)/15 = 1.67

Hence,
P(D>45) == P(Z>z)
= P(Z>1.67) = 1 - P(z<= 1.67)
= 1 - 0.9525 [From Normal table]
= 0.0475 ...

Solution Summary

A problem related to blood supply and demand/requirement is solved, for different possible probabilities.

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