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I am teaching 15 year olds basic probability. I am teaching them "experimental probabilities" (eg the probability that it will rain on a given day in August is 0.15) vs "theoretical probabilities" (eg the probability of throwing 3 Heads in a row is 1/8).

When I asked them for some examples of experimental probabilities, a couple mentioned one-off events, like the probability that their team wins their next match, or that it rains tomorrow.

I can't see that one-off events can have a probability other than 0 or 1. They cannot be re-run and hence their is no basis for determining or assigning a probability.

I would like to correct my students if and when they next make this error, but before doing so I would like to make sure of my facts. Is it legitimate, for example, to talk about a 40% chance of rain tomorrow, and if so, what does it mean?

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It depends on how you think about probability. If you are a frequentist, which is implied by the way you talk about one-off events, then it doesn't really make sense. Or you have to think about them in some very convoluted way. But if you have a Bayesian look on probability, then probability reflects your subjective assessment of the outcome of an event. And in that case, one-off events can be given a probability. – Raskolnikov Jun 11 '13 at 13:16

40% chance of rain tomorrow usually means (Laplace's definition) that on average, with the same meteorological conditions as you observe today, it will on average rain 40% of the time.

That you cannot repeat tomorrow, does not mean that you cannot repeat the same conditions as the ones in which tomorrow will happen.

Where this does reflect, is in the state of the random variable. If you want the random variable for it raining tomorrow or not, it will be $0$ or $1$ integer, not a real number.

Note this is not true with a baseball game, for example, since it may rain out and some have been known to end in a draw (I remember an All-Star game like that), although the chances for either one should be pretty thin...

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When you assign probabilities to real-world events, you almost always idealize things slightly to obtain some repeatable expirement which you can then assign probabilities too. Sometimes, like if you ask "what's the probability of throwing a three with a fair dice", the idealization is pretty natural - it certainly seems sensible to treat throwing a dice as an repeatable experiment.

If you ask "what's the probability that it will rain tomorrow", the kind of idealization you have in mind isn't immediatly obvious. You could be referring to the date, and effectively be asking "what's the probability of rain on a certain date". Or, you could be asking "what's the probability of rain giving the current state of the earth's atmosphere". Or "what's the probability of rain given that fact that the weather forecast was sunny".

So the question isn't so much whether an event is one-off or not - all real events are, strictly speaking, one-off - but whether there a sensible idealization in which the event is repeatable, and whether that idealization is obvious enough to not require an explanation.

If I was in your place, I'd tell my students that when they say "What's the probability of rain tomorrow" that they have to explain the idealization they're assuming, i.e. instead say maybe "What's the probability of rain tomorrow given that ....".

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Chris Burdzy explored this question in great detail in this book.
He essentially argues that it's fine to assign a probability to a one off event, and if you can't then probability theory is a pretty useless tool. (His argument is obviously more complex than that.)

I'm not sure whether I share his view or not, but it's a bit harsh to judge your students as wrong for agreeing with him.

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Probabilities of one off items are best computed using Dirichlet distributions and other multinomial distributions. The answer for "whether it will rain tomorrow" assuming it is summer is clearly not 0, because there is always a small chance it will. Here is a write up which I think you may find useful.

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