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We consider how to make probability forecasts of binary labels. Our main mathematical result is that for any continuous gambling strategy used for detecting disagreement between the forecasts and the actual labels, there exists a forecasting strategy whose forecasts are ideal as far as this gambling strategy is concerned. A forecasting strategy obtained in this way from a gambling strategy demonstrating a strong law of large numbers is simplified and studied empirically.arXiv:cs/0505083v1 fatcat:x7yaamqir5e4hews3iorhwwxoa