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"x% chance of rain" means they predict x% of the time it will be raining in any given spot in the given area over the given time period.
Now that we have that sorted out, just like weather probabilities, polls have a margin of error. It's a mistake to assume experts are predicting an exact outcome like 20% chance of rain or Hillary will beat Trump by 2% or whatever.
To be "wrong" in the scientific sense, it's generally agreed an outcome has to be outside a 95% confidence interval (usually expressed in layman's terms as a "margin of error"). That would be an outcome so rare had the model been correct as to make one reject the validity of the model. Neither the Trump vs. Hillary election or the Brexit vote fell outside that margin of error, fyi.
In a US national election, predicting the outcome based on the polls is complicated by the fact that it's 50 separate races for president, with varying numbers of electoral votes for each state. Unless one person is comfortably ahead in a number of key states, it'd actually be quite common for a number of swing states to go against the polls but still be within the margin of error for that state, and with fifty states at play, on average 2.5 states would be expected to go outside the margin of error (1/20 chance per state). And if those states are critical ones (e.g., Florida), that can shift the whole outcome.
When people like Nate Silver say "54% chance of candidate X winning POTUS," or whatever, and the numbers go up or down as the votes get counted, what they are doing is modelling the outcome for each state as a separate probability, updating that information based on votes counted, and saying "within our model, with its predictions based on polls and modelling of unknown factors (input into the model as random noise), we expect that 54% of the time this candidate will win." The closer that number is to 50% the more they are admitting they don't have enough information to make a clear prediction.
Meteorologists use models based on current weather patterns to make their predictions, but rather than predicting a binary outcome event (X vs. Y wins an election) they are predicting an overall probability of an event, in a given area, over a given period of time.
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