The null hypothesis in assessing earthquake predictions is often, loosely speaking, that the successful predictions are chance coincidences. To make this more precise requires specifying a chance model for the predictions and/or the seismicity. The null hypothesis tends to be rejected not only when the predictions have merit, but also when the chance model is inappropriate. In one standard approach, the seismicity is taken to be random and the predictions are held fixed. ‘Conditioning’ on the predictions this way tends to reject the null hypothesis even when it is true, if the predictions depend on the seismicity history. An approach that seems less likely to yield erroneous conclusions is to compare the predictions with the predictions of a ‘sensible’ random prediction algorithm that uses seismicity up to time t to predict what will happen after time t. The null hypothesis is then that the predictions are no better than those of the random algorithm. Significance levels can be assigned to this test in a more satisfactory way, because the distribution of the success rate of the random predictions is under our control. Failure to reject the null hypothesis indicates that there is no evidence that any extra-seismic information the predictor uses (electrical signals for example) helps to predict earthquakes.