1 Uncertainty is not just a matter of meteorological ignorance. The new ideas of chaos and nonlinear dynamic imply predictability limits even with perfect knowledge of the physics. That perfect knowledge would yield perfect forecasts only in the case where we have infinitely accurate measurements of every relevant quantity everywhere all the time. Our current observational system is rather short of that ideal.
2 Under these conditions, the m categories yi are said to be mutually exclusive and exhaustive.
3 A thunderstorm's probability is conditional on the simultaneous presence of moisture, instability, and lift.
4 There are other viewpoints. This will be explained as we consider verification.
5 This leads to the question of how to interpret the values inside the contours. It's possible to assume that all the pseudo-points that are touched by the contour have the value assigned to the contour, but what about in between contours? One possibility is to interpolate values to all the interior pseudo-points. Another is to assign the contour value to everything inside a given contour up to the next interior contour. The latter means that only a limited set of probability values are allowed. Yet another interpretation is that probabilities are binned, such that everything inside one contour and exterior to another falls in a particular assigned bin range. The choice among these and any other possible interpretations is somewhat arbitrary and depends on the verification being done.
6 That is, it contains the totals for the ensemble of forecasts, which might cover many days and many forecast periods. To examine the time-dependendent information, you must reconsider the individual forecasts.