Is it correct to compare probability percentages by their ratio?

Question

Imagine we have 2 sets. Each has a probability of a certain event. It is 0.01% for the set one and 0.02% for the set two.

Is it correct to compare these probabilities by their ratio? That is saying, that the second set has twice the probability of the event compared to the first one.

My problem with this is that increase in probability from 0.01% to 0.02% is so negligible, so that saying it is twice more probable looks sooo misleading to me.

How does answer to this question change if we know both sets to be representing the same amount of trials?

How does answer to this question change if we do not know for certain whether sets are of equal amount of trials?

I have found this question, but I believe it does not apply here.

Answer 1

You can compare probabilities by ratio like this, but you have to know how to interpret it.

I think a really good "paradox" that precisely captures intuition difference is the "potato paradox".

It says:

Fred brings home 100 pounds of potatoes, which (being purely mathematical potatoes) consist of 99 percent water. He then leaves them outside overnight so that they consist of 98 percent water. What is their new weight? The surprising answer is 50 pounds.

The idea here is that only $1$ pound out of $100$ was "potato" matter on the first day, but since the second day, it was $2\%$, and you can't magically have more or less "potato" matter, we can conclude that now half the water dried up, so now only $1$ pound out of $50$ is "potato" matter and hence it's $98\%$ water.

The idea here is that it's misleading: you don't magically have more potato matter as it might initially seem. You just have less water. So in an almost exactly similar sense, if you have the same amount of positive outcomes in a smaller sample space, your probability would be higher.

So I suppose here's how I think about it: either the sample space is different, the number of outcomes is different, or both, when comparing probabilities.

If we know two different probabilities have the same sample space, like if we say disease rates in group $A$ are $0.01\%$ and disease rates in group $B$ are $0.02\%$ and group $A$ and $B$ are the same size, then we know twice the amount of people in group $B$ got sick compared to group $A$. There are two times as many sick people.

If we know two different probabilities have the same number of outcomes but not the same sample space, like with the previous example, if we know the same number of people are sick in group $A$ and group $B$ and the probability of the disease in group $B$ is twice as high, then we know there's half the number of people in group $B$. Group $B$ is half the size of group $A$.

If we know neither sample space size nor outcomes, then, like you said, we're comparing ratios of ratios. So it really doesn't tell you either the difference in sample space size or number of positive outcomes.

I think you'd find this video on Bayesian statistics, and similar videos/problems very interesting.

Answer 2

The percentages are small but one shouldn't let perception of size effect a mathematical reality. Given a large enough population, these can be quite different in size. Example: $.01\%$ of the US population will eventually die of disease A and $.02\%$ of disease B. The two numbers are $32 700$ and $65 400$.

I have another interesting related situation about perception of size. A person is purchasing a $\$30$ item at a store and is told they can get it for $\$20$ if they drive to their other branch a few miles away. Most people would drive the extra distance to save some money. Now, if they had been purchasing a $\$500$ item and were told they could get it for $\$490$ by driving to the other store, most people would not do it.

People equate cost benefit as a percentage but the reality of it is, and the real question is, would you drive a few miles to save $10 and whatever you are purchasing shouldn't (but does) come into it.

The bottom line is, one probability is twice as big as the other and in the case of the purchase, the saving was exactly the same.