For weather forecasts on TV, many claims are made about the probability of single events like there is a “40% chance of thunderstorms tomorrow.” Can claims like this make sense given the definition of probability as a ratio of favorable outcomes to possible outcomes? Is there another way of thinking about probability that help these claims make more sense?
- As Dave pointed out, one interpretation of the "40% chance of thunderstorms" is "historically, 40% of the days where the conditions were similar to Today, the next day there was a thunderstorm". This is a frequentist interpretation based on the interpretation of probabilities as frequencies. Even in the case where we don't have historical data or multiple realizations of the experiment, it is still possible to have a frequentist interpretation of probabilities. In such a case the statement "There's a 40% chance of X" is interpreted as meaning "If we were to perform this experiment multiple times, 40% of the outcomes would be X".
- A second interpretation is the Bayesian or subjective interpretation of probabilities: The probability represents a degree of likelihood, not based on number of outcomes, but based on our theoretical knowledge of the situation. Consider a roulette wheel, where half of the squares are red and half are black: You assign a 50% chance of the ball landing on black, not based on number of outcomes, but based on the geometry of the situation. Since half of the wheel is made of black squares and had is made of red squares, the likelihoods of the ball landing on either side are equal. In your weather forecast, a meteorologist might assign a 40% chance of thunderstorms because of their knowledge of hydrodynamics and of wind patterns, not based on any historical data. You will thus hear Bayesians speak of "A priori" probability, that is probability obtained from a theoretical model a situation, vs "A posteriori" probability, which is obtained from running the experiment multiple times.
IMO, the most pragmatic interpretation is that the implied assertion is that 40% of the times that forecasts predict a 40% chance of something, it happens.
For various reasons, you may try to break this down further; e.g. restricted just to forecasts of thunderstorms. Or appropriately incorporate other things (e.g. the frequency that 30% predictions come true)
The relevant ensemble is "the set of days where, on the preceding day, the weather forecaster assigned a 40% chance of thunderstorms" -- if the weather forecaster is accurate (approximately) 40% of those days will have had thunderstorms.
This is probably easier in terms of a bounded domain. Take a specified year, on each day record the forecaster's prediction, and the weather conditions. At the end of that year to can pick out the days where forecaster said "40% chance of thunderstorms tomorrow", and the set of days that followed those; That set is your population from which you can make frequentist claims about the probability.