Increasing concentrations of atmospheric carbon dioxide will almost certainly lead to changes in global mean climate. But because--by definition--extreme events are rare, it is significantly more difficult to quantify the risk of extremes. Ensemble-based probabilistic predictions, as used in short- and medium-term forecasts of weather and climate, are more useful than deterministic forecasts using a 'best guess' scenario to address this sort of problem. Here we present a probabilistic analysis of 19 global climate model simulations with a generic binary decision model. We estimate that the probability of total boreal winter precipitation exceeding two standard deviations above normal will increase by a factor of five over parts of the UK over the next 100 years. We find similar increases in probability for the Asian monsoon region in boreal summer, with implications for flooding in Bangladesh. Further practical applications of our techniques would be helped by the use of larger ensembles (for a more complete sampling of model uncertainty) and a wider range of scenarios at a resolution adequate to analyse average-size river basins.