Question

question 4 of 4
during the initial months of the covid-19 pandemic, resources for testing samples for the disease were scarce. what if specimens were pooled and several samples were tested at once? if the pool tested negative, then all of the individuals who submitted specimens in the sample are free from the disease. if the pool tested positive, then at least one of the specimens in the pool was positive, and all of the specimens in the pool would be tested individually. in the long run, does this procedure result in fewer tests?
assume that 5% of specimens that are tested for covid-19 yield positive results. (the actual rate was much higher when testing was not extensive, but overall, when testing is being done extensively, 5% is a reasonable estimate.)
suppose that 1 million people present themselves for testing and that specimens are pooled together in groups of size 5. what is the expected number of tests that would have to be performed?
would pooling more than five specimens be useful? what is the expected total number of tests that would be performed if 10 specimens are pooled?
expected total number of tests:
tests

Explanation:

Step1: Calculate number of pools

There are 1,000,000 people, and each pool has 5 specimens. So the number of pools is $\frac{1000000}{5} = 200000$ pools. Each pool requires 1 initial test, so initial tests: $200000$ tests.

Step2: Calculate probability a pool is negative

The probability a single specimen is negative is $1 - 0.05 = 0.95$. For a pool of 5, the probability all are negative (so pool test is negative, no further tests) is $0.95^5 \approx 0.77378$.

Step3: Calculate probability a pool is positive

Probability a pool is positive is $1 - 0.77378 = 0.22622$. For positive pools, we test each of the 5 specimens individually, so each positive pool requires 5 additional tests.

Step4: Expected additional tests per pool

Expected additional tests per pool: $0.22622 \times 5 \approx 1.1311$.

Step5: Total expected tests

Total expected tests = initial tests + (expected additional tests per pool × number of pools)
Initial tests: $200000$
Expected additional tests: $1.1311 \times 200000 \approx 226220$
Total: $200000 + 226220 = 426220$? Wait, no—wait, initial test per pool is 1, then for positive pools, we do 5 more. Wait, maybe better:

Alternative approach:
For each pool:

• With probability $p = 0.95^5$, we do 1 test (pool test).
• With probability $1 - p$, we do $1 + 5 = 6$ tests (pool test + 5 individual tests).

So expected tests per pool: $p \times 1 + (1 - p) \times 6$
$p = 0.95^5 \approx 0.77378$
Expected per pool: $0.77378 \times 1 + 0.22622 \times 6 \approx 0.77378 + 1.35732 = 2.1311$

Then total expected tests: $200000 \times 2.1311 \approx 426220$? Wait, but wait—original total if no pooling: 1,000,000 tests. But with pooling, let's check again.

Wait, number of pools: $N = 1000000 / 5 = 200000$ pools.

For each pool:

• If all 5 are negative (prob $0.95^5$), we do 1 test (the pool test).
• If at least one is positive (prob $1 - 0.95^5$), we do 1 (pool) + 5 (individual) = 6 tests.

So expected tests per pool: $E = 0.95^5 \times 1 + (1 - 0.95^5) \times 6$

Calculate $0.95^5$:
$0.95^2 = 0.9025$
$0.95^3 = 0.9025 \times 0.95 = 0.857375$
$0.95^4 = 0.857375 \times 0.95 = 0.81450625$
$0.95^5 = 0.81450625 \times 0.95 = 0.7737809375$

So $E = 0.7737809375 \times 1 + (1 - 0.7737809375) \times 6$
$= 0.7737809375 + 0.2262190625 \times 6$
$= 0.7737809375 + 1.357314375$
$= 2.1310953125$

Total expected tests: $200000 \times 2.1310953125 = 426219.0625 \approx 426219$ (or 426220 when rounded).

Wait, but let's verify. If we didn't pool, we'd do 1,000,000 tests. With pooling, we expect ~426k tests, which is less. But let's check the steps again.

Number of pools: 1e6 / 5 = 200,000. Each pool has 1 test, then for positive pools, 5 more. The probability a pool is positive is 1 - (0.95)^5 ≈ 0.2262. So expected number of positive pools: 200,000 × 0.2262 ≈ 45,240 pools. Each positive pool requires 5 additional tests, so 45,240 × 5 = 226,200 additional tests. Plus the initial 200,000 pool tests: 200,000 + 226,200 = 426,200 tests. Yes, that matches.

Answer:

$\boxed{426220}$ (or approximately 426219, depending on rounding)