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It always feels wrong when I read something like "ABC is 3 times less likely than DEF", when it would make more sense to just say "ABC is a third as likely as DEF" or "DEF is 3 times more likely than ABC". Is the "3 times less" usage against any sort of math language rules, or just something I'll have to learn to live with?

PolymorphismPrince
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Brian Taylor
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    I agree with the OP that this is incorrect and unclear usage, and it annoys me as well. But it seems to be more of a linguistics issue than a mathematical one, and is therefore probably off-topic for this site. – mweiss Aug 05 '21 at 22:56
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    This is a question about language rather than about mathematics. Language means what the culture has decided it to mean. "P is x times less likely than Q" is a common expression that is used in everyday language, so yes, you should learn to just live with it. The point is... whether you choose to say "P is x times less likely than Q" or "Q is x times more likely than P"... it is common for the condition or group you are more interested in talking about is mentioned first. – JMoravitz Aug 05 '21 at 22:58
  • For instance... Unvaccinated people are three times more likely to catch Covid than Vaccinated people (where we include catching with no symptoms). This draws attention to the fact that the unvaccinated are what we are interested in... we are pointing out how being unvaccinated is more dangerous and they should get the vaccine. Compare to the statement "Vaccinated people are three times less likely to catch Covid than the unvaccinated" where here the focus is on the vaccinated and pointing out that it is good to have been vaccinated but 3 times is less than one might have thought. – JMoravitz Aug 05 '21 at 23:02
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    My peeve with this language is that it requires us to use “three times more” to mean what a mathematician would say is “three times.” If I say “50 percent more” I mean 150% total, so I’d expect “three times more” to mean 400%, but it just means 300%. Still, just get used to it, even if you hate it. Not much can been done about it other than not using it yourself. – Thomas Andrews Aug 05 '21 at 23:18
  • Why can't people just write "67% less likely that ABC"? – peterwhy Aug 06 '21 at 00:30

1 Answers1

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What does "three times less likely" mean? The only natural interpretation I can give to it is to interpret it as "three times as unlikely". That is: if we can measure how unlikely an event is, it makes sense to compare the two events by using a ratio of unlikeliness.

For example, suppose $A$ is an event with a probability of 80%, so its unlikeliness -- that it, the probability that it doesn't happen -- is 20%. If $B$ is a second event with a probability of 40%, then its unlikeliness would be measured at 60%.

Notice that in this example, if we focus on likeliness, we would say that $A$ is twice as likely to occur as $B$, or we could say that $B$ is half as likely as $A$. But if we focus on unlikeliness, we would say that $B$ is three times as unlikely as $A$.

That is to say: "$B$ is three times as unlikely as $A$" and "$A$ is three times as likely as $B$" are both meaningful statements, but they do not mean the same thing.

All of these are meaningful ways of describing reality, and depending on what you are interested in, you might choose one or the other. If you are trying to measure the risk of contracting an infectious disease, you might say that one population is at twice as much risk as another. If you are trying to measure how safe you are from an infectious disease, you might say that one population is three times safer than the other. (Personally I think the second formulation is less clear and more likely to be misunderstood, but it seems to be mathematically coherent, in this specific context.)

What's really wrong is when people use this verbal formulation to describe things that don't even make sense. "Three times smaller" is the example that grates on me the most. If building $A$ is 180 feet tall and building $B$ is 60 feet tall, it makes sense to say that $A$ is three times larger than $B$, or that $B$ is one-third the size of $A$, but "$B$ is three times smaller than $A$" doesn't make any sense at all. How do you measure "smallness" of an object? What is the reference point for "not at all small"? "Three times smaller" is not just unclear, it's incoherent.

mweiss
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    I’m not sure inventing your own meaning is a useful answer. For what it is worth, “A is three times less than B” as actually used is the same as “B is three times A.” – Thomas Andrews Aug 05 '21 at 23:59