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How would you formalize the statement: "The vote tally for Gov. Tush was within one-hundredth of one percent of one-half the total number of votes cast"?

For example, the answer key claims the correct formalization is:

(0.9999) * (total votes cast / 2) <= (Tush votes) <= (1.0001) * (total votes cast / 2)

But this formalization seems to be saying something different than the original statement. This formalization seems to be saying that "Tush votes" were within 99% of half the votes cast and 100.01% of half the votes cast.

I interpret the statement as saying Tush has received at most one-hundredth of one percent of one-half of the total votes cast. If his tally is within one-hundredth of one percent of half the votes cast then wouldn't that look like this:

0 <= (Tush votes) <= (.0001) * (total votes cast / 2)

My formalization looks like the mathematical definition of the statement (notice that the Tush votes are between the comparison operators, hence within, correct?)

From what I've written here, does it seem like I am misinterpreting something?

Matthew S.
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    I agree with the answer key. That's my understanding of the English sentence. I would render your interpretation something like, "The tally for Gov. Tush was no more than one one-hundredth of half the votes cast." – saulspatz Aug 26 '20 at 02:29
  • How about $$ \bigg | \frac{2x}T-1 \bigg|<.001$$ – WW1 Aug 26 '20 at 02:41
  • Yes my formalization is basically saying that. But what do you think about my interpretation of the answer key as stated in my question? I interpret the answer key to be saying ""Tush votes" were within 99% of half the votes cast and 100.01% of half the votes cast." The other thing I don't get about the prescribed formalization is that it uses the number 1.0001 which is over 100% – Matthew S. Aug 26 '20 at 02:59

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I think the English sentence is saying that the votes received were "within a certain margin of error of some value," i. e., that "'Tush votes' were within 99% of half the votes cast and 100.01% of half the votes cast" as you say. This is also what the user WW1 means in his comment.

Perhaps this seems strange to you, but there are actually several methods of counting votes, each with different rates of error (accuracy). Increasing the accuracy may also come at a cost since it may be more labor intensive and so on, so counting votes is more complicated than it seems a priori.

So here we are saying that after counting the votes, the tally for Gov. Tush was half of the votes, with an accuracy of 0.01%.

Note also that there is a marked difference between being "x percent of y" - essentially your formalization - and being "within x percent of y" - which is what the problem sentence is saying.

Moreover, in this case the number $1.0001$ being "over 100%" is not really a problem; it is just a multiplier that gives you a number slightly over half the votes. Now if the received votes were to exceed the total number of votes, that could be a problem, since it might be an indication of an error in the counting process (of course, after due consideration of the accuracy of the method) or possibly cheating.

Hope this helps to sort things out :)

AntimatterHedgehog
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