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In the probability thread, multiple philosophers mention examples of zero-probability events that aren't necessarily "impossible" (like flipping an infinite number of heads in a row). Arriving at a probability of zero in these instances relies on saying that 1/infinity = 0. But this math seems misleading. Don't mathematicians rely on more precise language to avoid this paradoxical result, by saying that "the limit of 1/x as x approaches infinity = 0," rather than simply "1/x = 0"? I feel like there must be some way to distinguish (supposedly) zero-probability events that are actually possible and zero-probability events that are impossible. Thanks!

To answer this question, it may be helpful to say something about the mathematical formalism usually used in probability theory. The first step in applying probability theory to study some random process is to identify the set of all possible outcomes of the process, which is called the sample space . For example, in the case of an infinite sequence of coin flips, the sample space is the set of all infinite sequences of H's and T's (representing heads and tails). Probabilities are assigned to events , which are represented by subsets of the sample space. For example, in the case of an infinite sequence of coin flips, the set of all HT-sequences that start with H represents the event that the first coin flip was a heads, and (assuming the coin is fair) this event would have probability 1/2. The set of sequences that start with HT is a subset of the first one, and it represents the event that the first flip was heads and the second tails; it has probability 1/4. Now, consider some infinite HT...

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