Is "30 times more likely" equivalent to a "3000% greater probability"?

Question

I am trying to make a persuasive point based on facts and would like to be most accurate / clear in my point. Would "30 times more likely" be equivalent to a 3000% greater probability"

Answer 1

The percentage analysis is the same for probabilities as anything else: $30x$ is $3000\%$ of $x$, but $2900\%$ greater than $x$ if $x>0$.

Answer 2

If event $X$ is "$30$ times more likely than event $Y$", then if the probability of $X$ is $x$, then the probability of $Y$ is $30x$ or, what is the same thing, $x + 29x$ ... which is $2900$% of $x$ more than $x$

That is, if, say, $x=1$%, then $y$ would be $30$%

So, $y$ is not $3000$% of $x$ more than $x$ ... for then $y$ would be $31$%

And, $y$ is certainly not $3000$% in addition to $x$, for then $y$ would be $3001$% ... which makes no sense.

You have to be especially careful of this latter difference in which people may talk about relative percentages. For example, a bank may say that they can offer you a loan with an "interest rate that is $1$% lower than our main competitor!"

OK, so what exactly does that mean? You might think that if the competitor offers a loan with a $10$% interest rate, then this bank offers one with only a $9$% interest rate, right? Except that maybe what they mean is that it is $1$% of that interest rate of $10$%, which would only be $0.1$%, i.e. their interest rate is $9.9$%. Still better for you, of course, but not as good as you might have thought!

Answer 3

On the language side of this: I think phrases like thirty times more likely should be avoided and replaced with thirty times as likely. The reason for my fussiness is easier to see for smaller numbers.

Suppose you do something once, then do it three times more. You've done it four times. So logically, something that's three times more likely ought to be four times as likely.

Mostly, everyone understands that the words are being used loosely and treats $n$ times more and $n$ times as as synonyms. But I think this still contributes to confused thinking about numbers and statistics, because the literal meaning of the words conflicts with the intended mathematical meaning. The words used ought really to be ones which help people follow the maths if they want to. That is, someone who looks more closely at the words should find that they're seeing the maths more clearly as well.

Here's an example where the phrase really is ambiguous: I've done the shopping three times more than you. Does this mean "three more times", or "three times as often"?

Incidentally, the slip you made is exactly the kind of confusion I mean: you added thirty times the original to account for the more, ending up with $31$ times the original. Maybe thirty times as would have made it easier to think clearly and avoid the mistake.

And as others have said, $30$ times as likely is $2900\%$ more likely, because $30$ is $29$ more than $1$.