Question 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!

Accepted:
June 25, 2009

Comments

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-sequence s. For any positive integer n, we can consider the set of all sequences that agree with s for the first n terms. This set contains s, and imitating the reasoning in the last paragraph we see that it represents the event that the first n coin flips come out as specified by s, which has probability 1/2n. Since {s} is a subset of every one of these sets, the event that the entire infinite sequence is exactly s must have probability less than 1/2n for every n. But that means that the event must have probability 0. So you are absolutely right that the reasoning here involves a limiting process: the probability is 0 because 1/2n approaches 0 as n approaches infinity.

With this background, it is also now easy to see the distinction between zero-probability events that are possible and those that are impossible. The event that the entire infinite sequence is s is represented by the set {s}. It has probability 0, but is possible. The event that the first flip is both a heads and also a tails is represented by the empty set (since there are no elements of the sample space that fit this description); it has probability 0 and is impossible.