One thing that is fascinating in weather prediction by computer models (i.e., Numerical Weather Prediction - NWP) is the tremendous amount of uncertainty that is inherent to the system. The basic concept of computer forecast models is first starting with an initial condition - i.e., a snapshot of the atmosphere at a given time. This snapshot is taken by combining observed atmospheric data (e.g., satellite, radar, upper air soundings) with a first guess of what the atmosphere "should" look like at that time based on a model that run earlier. This snapshot is then projected forward in time using mathematical equations that - with a balance of accuracy and efficiency - represent the physical, chemical, and thermodynamic processes that regulate the changes of the earth/atmosphere system.
Unfolding this scenario helps to explain the uncertainty. First, the initial conditions are subject to error because the atmosphere is very poorly sampled. For example, weather balloons carrying instruments to sample the upper atmosphere are launched hundreds of miles apart every 12 hours. Second, any instrument used to measure the atmosphere has intrinsic error. Third, the mathematical equations that represent all the processes take several shortcuts that only approximate how the atmosphere evolves - otherwise, it would take way too long to perform all the calculations and get results in a timely manner. Fourth, our physical understanding of the atmosphere is limited, which make it even more difficult to model with fidelity. Fifth, and most interesting of all, the atmosphere is a system that is subject to the butterfly effect, which is a sensitivity to the initial conditions. A slight change to the initial conditions, even well within the expected range of error introduced by the limitations mentioned above, leads to completely different forecasts - especially further out in time.
Any single computer model weather prediction is a deterministic system - that is, it will always give the same results with the same initial conditions. Unfortunately, it can't be known a priori how accurate the initial conditions are, or how sensitive sensitive the forecast is to slight changes in the initial conditions - changes that in fact fall well within the error of the initial conditions. In other words, there is no measure of the uncertainty associated with a forecast. There only is a single forecast - like, say, "the high temperature will be 54 degrees on Thursday" without any mention of confidence or other possible outcomes (e.g., "there is a 70 percent chance the high temperature Thursday will be between 52 and 57 degrees"). The inherent uncertainty for a particular forecast can be estimated by running several computer models with slightly different initial conditions and/or computing methods and comparing the results. If the resulting forecasts differ greatly, then it can be assumed there is more uncertainty and less confidence with that particular forecast. If the resulting forecasts are similar, then greater confidence can be assigned to the forecast.
So, weather forecasts are best treated probabilistically since all of these uncertainties dominate. The same applies to other modeled systems like economics, traffic flow patterns, etc. BUT, this does not mean that any of these systems is driven randomly. We assign probability to that which we are unable to observe in sufficient detail. Take the Heisenberg uncertainty principle , which postulates that there are systems where increasing the precision in measuring one variable decreases the precision with which another variable can be known. For example, the more precisely we determine the position of an electron, the less we can know its momentum. The electron HAS a specific position and momentum all the time, but we can't know these things simultaneously given our limited powers of observation. With a sufficiently precise measurements of the initial conditions and a sufficient understanding of the system, we could know exactly where and when the marble will fall into the roulette wheel - but again, we're limited by an insufficient abiltity to observe and understand the system. Is randomness somethings that can force a certain outcome then? As R.C. Sproul eloquently described in Not a Chance: The Myth of Chance in Modern Science and Cosmology, chance doesn't cause anything. It only attempts to quantify uncertainty that stems from our limited powers of observation and understanding. The idea of chance being a causative agent is a huge part of the argument for evolution, by the way, but I'll save that for another post.
