The models used to predict a trend are not the same models used to predict a short term result.
This is correct.
Predicting the result of a series of coin flips involves simple probability, and gets more accurate the bigger the sample size.
Predicting the result of a single coin flip with any degree of accuracy would involve analysis of a lot of factors that average out over the long term.
The difference of course, is that in a series of coin flips, each individual flip is an independent event. The outcome of previous flips has no bearing whatsoever on future flips. Consequently, over time, average probabilities are borne out by the increasingly large data set so that long-term predictions become more accurate.
With both weather and climate modeling, future conditions are 100% dependent on previous conditions, which are modified according to the models' algorithms before being fed back into the model as new input. Consequently, errors are magnified over time by both real-world environmental feedback and model-induced feedback, such that long-term predictions become less accurate.
Yes - the butterfly effect. But not just the butterfly in the real world - also the algorithmic butterflies.
<EDIT>Also, in your coin flip analogy, we know the probabilities beforehand. We know that the coin is 50% likely to land on heads and 50% likely to land on tails. So a long term prediction of 10,000 flips will be more accurate than a short term prediction of 10 flips. With weather and climate modeling, we don't know the probabilities beforehand. The probabilities are in fact what we're predicting: 30% probability of rain or 70% likelihood of warmer temperatures. So a long term prediction of the probability will not be confirmed simply because we've added more data points. In fact, because the climate will be impacted by the feedback mechanisms discussed above, it is far more likely that our short term prediction of a probability will have more accuracy than the long term prediction.</EDIT>
Still, your point about climate models being different than weather models is accurate. Climate models are engineered for lower resolution in almost every way - lower spacial resolution, lower temporal resolution, fewer independent meteorological parameters. In this way, modelers hope to limit the opportunity for compounding the errors (discussed above - and there are
always errors to compound) in order to push the time horizon much further down the road so that they can make a guess at what general conditions will look like say 10, 50, 100 years from now without the output getting swamped by the feedback.
Weather models predict specifics for tomorrow. Climate models predict generalities for next decade. That's the difference. And there's no statistical evidence suggesting increasing accuracy over the long haul, as your inaccurate coin flip analogy suggests. To the contrary,
every point of evidence indicates rapidly decreasing accuracy over increasing iterations (and consequently feedback).