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Problem 42 Medium Difficulty

Blood testing Suppose that a blood test for a disease must be given to a population of $N$ people, where $N$ is large. At most $N$ individual blood tests must be done. The following strategy reduces the number of tests. Suppose 100 people are selected from the population and their blood samples are pooled. One test determines whether any of the 100 people test positive. If the test is positive, those 100 people are tested individually, making 101 tests necessary. However, if the pooled sample tests negative, then 100 people have been tested with one test. This procedure is then repeated. Probability theory shows that if the group size is $x$ (for example, $x=100$, as described here), then the average number of blood tests required to test $N$ people is $N\left(1-q^{x}+1 / x\right),$ where $q$ is the probability that any one person tests negative. What group size $x$ minimizes the average number of tests in the case that $N=10,000$ and $q=0.95 ?$ Assume that $x$ is a nonnegative real number.

Answer

X=5

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Video Transcript

Okay. So for this problem, they give us an equation of 10,000 times one minus 0.95 to the X power most one over X. And they just want to know what X gives us the minimum value. So to minimise this s o the average equals that this is our objective equation. And we only have one variable, so we don't really need constraints here. So we take the derivative. It'll have the same zero if we drop this 10,000 or not. Eso The derivative is just negative. Natural log of 0.95 at times 0.95 to the X and that minus one over x squared. Okay, In lieu of trying to solve this algebraic lee, I just grafted to see where it crosses the X axis. Um, so just be careful in your in your calculators. Um, make sure you put parentheses around that natural auger pregnancies around the 0.95 just to make sure it doesn't evaluate Natural. Got the whole thing. Sometimes calculators can give a hiccup with natural log. Um, it crosses the X axis somewhere close to fot 5.2 And since the closest group size to 5.2 is five. Then get with the group size of five to have the minimum number of tests, and that's all, thank you very much.

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