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August 20th, 2008, 11:22 PM   #1
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Probability and Algebra problem

There has been an outbreak of a rare but fatal infectious disease nationwide. This incidence rate for a large population is 1 in 25000. Assume the incidence rate is uniform across the population.

The public health department wants to test every individual in the population for the disease. This can be done with a small blood sample from each member of the population.

Assume the population to be tested consists of 1 million people, and that each blood test costs $10. Then the total cost of testing the population is $10 million.

A cheaper way to test every individual is to combine a certain number of blood samples (into a "cluster") and to administer just one test to this cluster. For example, assume that 1000 samples are combined, so the "cluster size" equals 1000. The blood test is administered just once to the entire cluster. If the test is negative then the outcome is that none of the 1000 individuals are infected. For $10, 1000 people have been tested.

If the test is positive, each individual in the infected cluster will be re-tested.

A clever mathematician maintains that for the population, incedence rate and cost for the blood test given above, the average cost of administering the blood test for a cluster of 1000 is well below half a million dollars.
Algebraically show that this is true.

Any help with this would be greatly appreciated.
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August 21st, 2008, 04:15 AM   #2
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Re: Probability and Algebra problem

1000 initial tests @ $10 each --> $10,000.
There are 1,000,000 / 25,000 = 40 people expected to be infected.
At most there will be 40 infected people per infected group; at least 1. Thus there are 1 to 40 infected groups, requiring 1,000 to 40,000 additional tests.
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August 21st, 2008, 01:44 PM   #3
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Re: Probability and Algebra problem

Thanks for the reply, I will continue working this question with the information you have given me.
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