Daily logic 6

$2000$ actually have the disease. $(2$ % $×10,00,000)$

As false negative rate is $1$ %, Out of these $2,000$ , $\frac{1}{100}×2,000=20$ will be detected negative.

So, only $1,980$ will be detected positive.

The remaining unchecked people are $10,00,000-2,000=9,98,000$

As false positive rate is $2$ %, $\frac{2}{100}×9,98,000=19,960$ will be detected positive.

Total positive $=19,960+1,980=\boxed{21,940}$

False Negative & False Positive This terminology is used in all sorts of things that give only two possible outcomes, spam filters, virus tests (computer,) and what we are doing here.

False Positive simply is when someone is given a positive result that shouldn't i.e. the person does not have the virus False Negative conversely is when someone is given a negative result when they shouldn't i.e. the person has the disease.

Thus, perhaps not completely obvious, the True Positive rate is 1 - (False Negative rate) and True Negative rate is 1 - (False Positive rate)... Consider if you are testing 100 people that have the disease, there can be no false positives but you can still get some false negatives. In our case, the false negative rate is 1% or 0.01, so when testing 100 infected people we would get results indicating 1 negative and 99 positives. So the True Positive rate is 99%. Another way of phrasing the question, an alternative nomenclature, would be that the test is 99% accurate for positive results and 98% accurate for negative results.

People often concern themselves with false negatives, for example how much spam gets through your spam filter. But really the damage is done by the false positive. Real mail being blocked. Back to the Kingdom We'll work with 1,000,000 people, it's slightly easier than working with the percentages all the time. Let's calculate the number of people that will be found positive. Firstly the True Positives: population * infected rate * (1 - FNr) 1,000,000 / 500 * 0.99 = 1,980

Now the False Positives population * (1-infected rate) * FPr 1,000,000 * 499 / 500 * .02 = 19,960

We can see what is happening the True Negative rate might be 98% is high but the remaining 2% is acting on the huge number of people who are not infected. X is 1,980 + 19,960 = 21,940 Y is 21,940 / 1,980 = 11 ish= 11.08080808...

Number of Phagers
= (1/500) × 1000000
= 2000

Number of Phagers tested negative
= 1% × 2000
= 20

Number of non-Phagers tested positive
= 2% × (1000000 - 2000)
= 2% × 998000
= 19960

King's X
= (2000 - 20) + (19960)
= 21940

Chief Nerd's Y
= 21940 / (2000 - 20)
= 21940 / 1980
= 11.0808081
≈ 11