Events that are impossible have zero probability, but the converse is not necessarily true. Clearly an empty set has zero probability. But, a zero probability event does not mean an impossible event.
The simplest example comes comes from a continuous model. Every point has zero probability but every point can be a possible outcome. Similarly, in a 2-dimensional space, a line also has zero probability. In other words, only events that have a significant enough likelihood can have positive probability in a continuous space.
To take a more real example, consider an infinite number of fair coin flips. The probability of seeing all heads is zero! But, it is a possible outcome.
This reminds me of a tongue-in-cheek note that I wrote to a friend, Ray, on his chances of finding the right girl, by his definition. The analysis is posted below, along with a postscript.
The numbers and percentages used were from sources dating back to 2008 and may no longer be appropriate to use. Please note that this study was the result of about twenty minutes of procrastination and is not a serious endeavor to quantify something as elusive as meeting that special someone.
1,907: must be smart
Total female graduate students at MIT — assume all graduate students at MIT are in Ray’s age group and smart — a similar analysis can be performed using a different cohort but I expect it would indicate an equal or more pessimistic result
1,158: must be culturally similar (raised in the US)
Remove 39.3% who are international
135: must be culturally similar (ethnically Chinese)
11.7% are Asian — assume all Asians are Chinese
27: must love games
~20% are gamers — overestimate because most female gamers are older than 35, underestimate because we are considering Asians
12: must be nice and not overly religious
~45% are nice and are the non-delusional religious type — see notes below
0: must meet this person
0% Ray will actually go meet people — this is the most limiting factor
Final result: Ray is screwed.
45% is from rotten tomatoes’ rating of Constantine. Justification for why this is a reasonable assumption:
- People who voted this movie as fresh can appreciate humor that makes fun of religion and so are not overly religious, even if they are religious
- IMO, it is a nice person movie (subjective=expert opinion)
- The main actor is Keanu Reeves, so it takes out people who voted based on physical attraction (removes bias)