At the end of the probability concepts section, the major thing I want students to do is to interpret probability statements (which for some is the most difficult part of the course, never having encountered probability concepts before). I give a quiz question on the classic weather forecast precipitation probability: "Interpret this probability statement: 'There is a 40% chance of rain today in the New York area'". So personally, I always took this to mean that there was a 40% chance of getting any rain at all in New York today (40% chance for a drop of measurable rain somewhere in New York; i.e., over many days like today 40% of such days will get a drop of rain or more in New York).
But one of my students not only started researching this on her own, she actually called the New York weather service to ask a meteorologist how this was computed. She still didn't get the interpretation quite right (one of the few questions she missed all semester), but the discussion was enlightening for both of us.
The truth is that the weather forecast statement is in regards to rain at any random location in New York, not actually the rain for New York as a whole. I suppose that is really a more useful statement, after all. The publicized percentages are computed by multiplying the expected coverage area percent by the probability of rain occurring in that area (so if it's 40% likely that 100% of the area gets rain, you report the same result as an 80% chance that 50% of the ground gets rain). Therefore: What's being reported is the chance that any arbitrary point in New York gets measurable rain; i.e., 40% indicates that for any random point in New York, if we observe many days with conditions like today, 40% of such days get a measurable drop of rain on that point-location.
Links to more information:
- Comments from the National Weather Service, reposted at the University of Texas at Austin website: here.
- Video from a meterologist in Boulder, including citation to the 2005 Gigerenzer et. al. paper in Risk Analysis which surveyed people for their understanding of these statements (where I got my quiz question in the first place): here.