The paper shows that a least-biased estimate of the failure probability of a structural component or structure with random loads and strengths may be calculated by using the maximum entropy distribution. The method permits the probability of failure to be calculated directly from statistical load and strength data without the need to choose analytical distributions to represent the data. Comparative tests show that the MED-based method generates accurate failure probabilities. The sensitivity of failure probability values to changes in the statistical data is studied and is found to be very high. The fourth statistical moment of the data is found to be at least as important as the first and second moments in a sensitivity context. This implies that very accurate statistical data on load and strength is essential if confidence is to be placed in any calculated failure probability. Results show that inaccuracies in the data can lead to errors in failure probability values which can be measured in hundreds of percent and sometimes in orders of magnitude.