Discussion
Out of 1000 individuals in a town, 100 unidentified individuals are covid positive. Due to lack of adequate covid-testing kits, the health authorities of the town devised a strategy to identify these covid-positive individuals. The strategy is to:
(i) Collect saliva samples from all 1000 individuals and randomly group them into sets of 5.
(ii) Mix the samples within each set and test the mixed sample for covid.
(iii) If the test done in (ii) gives a negative result, then declare all the 5 individuals to be covid negative.
(iv) If the test done in (ii) gives a positive result, then all the 5 individuals are separately tested for covid.
Given this strategy, no more than _______ testing kits will be required to identify all the 100 covid positive individuals irrespective of how they are grouped.
Answer: A
No more than 700 kits are needed.
Step (ii) tests each of the \(1000/5 = 200\) pooled sets once: 200 kits. Step (iv) then tests the 5 members of every set whose pool is positive. The worst case is when the 100 positive people are spread one per set, making 100 positive pools:
\[ 200 + 100 \times 5 = 700 \]
Any grouping that puts two or more positives in the same set gives fewer positive pools and so needs fewer kits, which is why 700 is an upper bound 'irrespective of how they are grouped'. 600 assumes only 80 positive pools, 800 and 1000 overcount, and 1000 is what testing everyone individually would cost.
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