How Probable is an Infinite Sequence of Heads?
Williamson (Timothy)
Source: Analysis 67, 3 (2007): 173-180
Paper - Abstract

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Author’s Introduction

1. Isn’t probability 1 certainty? If the probability is objective, so is the certainty: whatever has chance 1 of occurring is certain to occur. Equivalently, whatever has chance 0 of occurring is certain not to occur (it has no chance of occurring). If the probability is subjective, so is the certainty: if you give credence 1 to an event, you are certain that it will occur. Equivalently, if you give credence 0 to an event, you are certain that it will not occur (it has no weight in your calculations of expected outcomes). And so on for other kinds of probability, such as evidential probability.
2. The formal analogue of this picture is the regularity constraint: a probability distribution over sets of possibilities is regular just in case it assigns probability 0 only to the null set, and therefore probability 1 only to the set of all possibilities. For convenience, restrict the term ‘possibility’ to those maximally specific in relevant respects. Thus possibilities are mutually exclusive and jointly exhaustive. The probability of a possibility is just the probability of its singleton. Assume that each possibility has a well-defined probability. Then regularity is equivalent to the constraint that every possibility has a probability greater than 0.
3. Regularity runs into notorious trouble when the set of possibilities is infinite, given the standard mathematics of probabilities, on which they are real numbers between 0 and 1. Indeed, when the set of possibilities is uncountable, no probability distribution is regular.

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