Unrepeatable experiments
In most scientific contexts, it is possible to imagine repeating an experiment in essentially the same way. However there are other scenarios in which such repetitions are clearly impossible, but there is still uncertainty about the outcome.
Examples
In these situations, the earlier definitions of probability cannot be used to describe uncertainty. Instead, a subjective or personal assessment of probability must be made.
Subjective probability can be given a formal definition, based on what you would regard as a 'fair bet' for the event. However we will not pursue subjective probabilities further in this book, largely because different individuals would have different subjective probabilities and hence may arrive at different, but reasonable, conclusions.
If either the equally-likely outcomes or the relative frequency definitions of the probability of an event can be used, the formal definition of subjective probability results in the same numerical value, provided it is applied by a 'reasonable' person with full knowledge of the experiment's structure.
Bayesian statistics
A whole field of statistics called Bayesian statistics is based on subjective probabilities and is gaining in importance, but it will not be considered further here.