What does "twice as likely" mean?

Question

Once in a while I hear people say something like

X is twice as likely as Y.

What they usually mean is:

$$p(X) = 2 \cdot p(Y)$$

and - in the context they refer to - they usually have $p(Y) < \frac{1}{2}$. But what do you do if $p(Y) > \frac{1}{2}$? Can there be an event $X$ that is twice as likely as $Y$? It also feels wrong to me to say that $p(X) = 100 \%$ is twice as likely as $p(Y) = 50\%$.

Is there a good definition what twice as likely means?

Some thoughts about this

Let's call this "twice as likely" a function

$$d: D \rightarrow [0, 1]$$

I would expect $d$ to have the following properties:

• $D = [0, m]\subseteq [0,1]$
• $d(0) = 0 $
• $d$ is monotonous

Answer 1

There's one interpretation which I think makes the most sense: $$p\longrightarrow p:1-p\longrightarrow 2p:1-p\longrightarrow \frac{2p}{1+p}$$ One could also interpret it as picking the best result out of $2$, in which case you would get $$p+(1-p)p=2p-p^2$$ Both are ${\sim}2p$ for low values of $p$.

I think the first is right, because then half as likely matches twice as unlikely. But if you generalize the second to positive reals, you won't get that picking the worst result out of $2$ is the same as being half as likely in the generalization.

Answer 2

Personally I don't think "double as likely as $p$" (or "twice as likely", which sounds more English?) sounds like it ought to be meaningful for $p\ge\frac12$.

However, if we want it to have a meaning, then it seems reasonable to require that "$q$ is double as likely as $p$" should mean $q=2p$ for small $p$ and $(1-q)=\frac12(1-p)$ for $p$ close to $1$. The problem is, of course, that these two expressions don't agree for $p$s in the mid-range, so we have to bridge them somehow.

Simply switching formulas at $p=\frac13$ -- which is the point where they give the same $q$ -- is insufferably simplistic, of course. It creates ugly discontinuities.

The most mathematically principled way to bridge the two ends would seem to be to say that $q$ should be the number whose logit is $\log2$ more than the logit of $q$. This gives us the equation $$ \log\frac{q}{1-q} = \log\frac{p}{1-p}+\log 2$$ which simplifies to $$ 2p-pq-q = 0 $$ or $$ q = \frac{2p}{p+1} $$ which is the same as user2345215 reached.

(This is, by the way, the rational function of lowest degree that has the required values and slopes at $p=0,1$).

Unfortunately, its results in the mid range are not quite intuitive. For example, it claims that "double as likely as" 33% ought to be 50% rather than the 66% one would probably expect.

Answer 3

I think separate "definitions" of "twice as likely" for $p < 1/2$ and $p > 1/2$ are both meaningful.

For $p < 1/2$, if you think of probability as successful outcomes over total outcomes, then a statement "twice as likely" means that the ratio of successful outcomes over total outcomes is doubled. Thus, it does not make sense to define the expression "twice as likely" if a probability is already more than $\frac{1}{2}$.

For $p > 1/2$, then interpret the statement as "an unsuccessful outcome is half as likely." Thus, we get that $99\%$ is "twice as likely" as $98\%$. Note that this is the same definition of the first except we are focusing on the other side of the coin, so to speak.

Answer 4

The 'people' you refer to who say "twice as likely" are not usually professional (or even recreational) mathematicians. Their use of this expression is therefore limited to instances where $P(Y) \leq 0.5$.

What 'people' mean when they say most things mathematical is usually the simplest thing it sounds like and which you've already stated in your question: To say that $X$ is twice as likely as $Y$ is to say that $P(X) = 2P(Y)$.

If $P(Y) > 0.5$, then nobody would ever say that some event $X$ is twice as likely as $Y$.

Of course, you are welcome to invent a "twice as likely" function with all the bells and whistles you like, but that will never be what 'people' mean.

Edit to elaborate: You add also that

It also feels wrong to me to say that p(X)=100% is twice as likely as p(Y)=50%.

I think it only 'feels wrong' because if $P(X)=1$ and $P(Y)=0.5$, then 'people' would typically say something (much) more emphatic like "$X$ will happen for sure, whereas $Y$ happens only half the time." The point of saying that $X$ is "twice as likely" as $Y$ is to emphasize that $X$ is much more likely to happen. But if $P(X)$ is going to happen for sure, you might as well simply say that it is going to happen for sure.