Probability zero
High school
Consider an event in a random experiment. The event has probability zero if its probability is $0$.
Undergraduate (probability)
Consider a probability space with probability measure $P$, and an event $E$. The event $E$ has probability zero if $P(E) = 0$.
Examples
Examples
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The probability that a uniform random number on $[0, 1]$ is rational
High school
The event that a uniform random number on $[0, 1]$ is rational has probability zero, because $\mathbb{Q} \cap [0, 1]$ has measure zero. The event is not impossible, however, because every rational number in $[0, 1]$ is a possible outcome.
Properties
- An event with probability zero can still happen. If $X$ is uniformly distributed on $[0, 1]$, then $P(X = 1/2) = 0$, but $X$ must take some value, and every value has probability zero.
Distinctions involving this term
- Types of set sparsity: Several terms describe a set that is “small” or “negligible,” even when it has infinitely many points. “Nowhere dense” means small in a topological sense (and a meager set is a countable union of nowhere dense sets), and “measure zero” means small in a size sense. Neither implies the other. An event of probability zero is a set of measure zero for a probability measure.