Think of it this way. Assume that a screening test for disease X costs $500 and finding it early averts $10,000 of costly treatment at a later stage. Are you saving money? Well, if one in 10 of those who are screened tests positive, society is saving $5,000. But if only one in 100 would get that disease, society is shelling out $40,000 more than it would without the preventive care.
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