Today, I want to address an issue with statements involving chance. To demonstrate, let’s first consider a statement that doesn’t involve chance:
“A cubic die tossed onto a flat surface will come to rest on one of its six sides.”
This claim can be empirically tested, with various dice and surfaces. If any one of our experiments results in the die spinning endlessly on a corner, we will have disproven the claim. We may have to refine the claim’s conditions; for instance, by requiring the presence of gravity. Nonetheless, it’s fairly clear what it means for the statement to be true or false. Now let’s try to make a claim involving probability:
“If a pair of standard dice are thrown, the probability of their face-up sides summing to nine will be one in nine (about 0.11 or 11%).”
What does it mean for this statement to be true? Unlike the first statement, this one doesn’t specify which result we’ll actually see. How can we possibly hope to test it, or to make use of its information?
The mathematician’s multiverse
Within the realm of abstract mathematics, we’re free to model probability in a way that fits our intuitions. Imagine a multiverse containing an infinity of possible worlds, whose total measure is 100%. Define the probability of an event, such as that of rolling a nine, to be the measure assigned to the subset of worlds in which the event actually occurs.