How can a possible event have probability zero?
Here is a paradox that confuses a lot of people when they first study probability:
The Setup
Suppose that you will arrive at a bus stop at a random time. The bus comes every 20 minutes, but because your time of arrival is (uniformly) random, then the time you have to wait for the bus is a uniformly random number from 0 to 20.
What is the probability that you will wait between 0 and 10 minutes?
Obvious because this is exactly half of all possible times, then the answer is 1/2.
Ok. Now what is the probability that you will wait between 5 and 10 minutes?
This is exactly 1/4 of all possible times, so 1/4.
Of course the process can go on. The probability of waiting between x and y minutes is going to be
\(\frac{y-x}{20}\)
The Paradox
take any specific amount of time. Say, 5 minutes.
What is the probability that your waiting time exactly 5 minutes?
Well, it’s
\(\frac{5-5}{20} = 0\)
That is to say, it’s a 0% probability.
Wait, but that seems strange. It is possible to wait exactly 5 minutes. And yet its probability is 0.
How can a possible event hat probability zero?
Resolution
There is no shared concept of “possible”.
First of all, we should note that the concept of “probability” in mathematics is rigorously defined and shared among all mathematicians.
On the other hand, “possible” does not enjoy such a common understanding. Mathematicians do not talk about what is “possible” in any rigorous sense (unless studying a very distinct subject called modal logic, which is a digression).
Therefore, in a formal mathematical sense, we do not actually agree that waiting 5 minutes for the bus is actually “possible”. More generally, mathematicians would resist saying that anything is possible—not because they don’t believe in possibilities, but only because they have not all agreed on what the word should mean. Mathematicians tend to insist on having rigorously defined terms first, and then discuss how these terms apply to examples.
A resolution that I reject: Possible means probability greater than 0.
Some people will argue: “possible” just means “probability greater than 0”. That’s intuitively an attractive resolution.
These people will then say “It feels like a paradox only because our intuitions about infinity are bad. We think that waiting for the bus for precisely 5 minutes is possible. But in a truly infinite space of events, it’s not actually possible.”
But I have a counter-argument. Clearly, in the intuitive sense, waiting precisely five minutes for the bus to arrive is possible. If you debate this, by saying
But it’ll never actually happen! It’s probability is zero. Whenever you wait for the bus, the actual waiting time will always be something like 5.000000107371… something like that!
Let’s imagine you’re right about that. Go ahead and wait for a bus. When it arrives, it will have arrived at some time. Let’s call that time \( t_0\). Whatever time that actually is, it is a precise real number, out to any place in the decimal expansion.
Because that waiting time actually happened therefore it is possible. (I am using the principle which I think almost everyone would agree to: Anything which actually happens can possibly happen. Nothing can be more possible than the actual!)
But now again ask the question: What is the probability that you wait precisely \(t_0\) time for the bus to arrive? Of course it is zero.
We admit every single time between 0 and 20 to be one of the waiting times that “could” happen. And yet every single one of them has probability zero. I think the only consistent use of the word “possible” requires us to accept each time as possible, and to accept that some possible events have probability zero.
Formal probability theory
In probability theory we have some common ideas:
There is the sample space, often denoted by \(\Omega\). This is the set of all “outcomes”, or whatever you want to call them. (Other words we sometimes use include “atoms” or “elements”.) In the bus example, \(\Omega\) is the set of all real numbers from 0 to 20, inclusive. Therefore \(0\in\Omega\) and \(\pi\in\Omega\) and so on.
There is the event space, often denoted by \(\mathcal F\). This is the set of all “events”. The events are things we can assign probabilities to. In the bus example, one example event is “waiting between 0 and 20 minutes”. Another event is “waiting between 5 and 10 minutes”. Another event is “waiting precisely 5 minutes”. Each event is a subset of \(\Omega\), whereas the outcomes are individual elements of \(\Omega\).
There is the probability function, often denoted by P. This is what actually does the job of assigning the probabilities to the events. In the bus example, recall the event of waiting between 0 and 20 minutes. Well it is the probability function P which assigns this event probability 50%.
Just for variety, you might want another example. Imagine a coin is tossed, so the sample space is the two events, \(\Omega = \{H,T\}\), corresponding to heads and tails. The events are: {H} meaning that heads occurs, {T} meaning that tails occurs, {H,T} meaning that either heads or tails occurs, and {} meaning that neither occurs. Now obviously the probability function will say that the probability of {} is zero, because the coin has to land some way. And it will say that {H,T} has probability 1, for the same reason.
The interesting question is what P says about {H}. If the probability of {H} is 2/3 then we say that \(P(\{H\})=2/3\).
The resolution.
Ok, with all of that formalism now briefly discussed, let’s get back to the question.
Here is my proposal: Every outcome in the sample space should be regarded as a “possible” outcome. That is to say, when we talk about what is “possible” we merely mean “what is in the sample space”.
I think this is intuitive because what we mean by the sample spaces is … well … just all the things that could possibly occur! If it couldn’t possibly occur we would not include it in the sample space.
This then implies that, yes, some events with probability zero are possible. It is a bit counter-intuitive, but I think it is a consistent use of these words.
And yeah, although it’s counter-intuitve, sometimes your intuitions just get things wrong. Intuitively, many people think one infinity can’t be bigger than another: But we know that’s not true in some sense. Intuition tells you that you shouldn’t be able to take apart a ball and then reassemble the pieces into two balls of equal size—and yet the Banach-Tarski paradox shows that one can.
So there you have it—waiting for exactly five minutes is both possible and has probability zero.