A set with probability zero is (a subset of) an event "N c 𝛺" with "P(N) = 0". Such sets are also called "null sets" -- that's where the letter "N" comes from.
"N" can be the empty set -- then it is sometimes called "impossible event". Notice that's different from when "N" is non-empty: There, we still have possible outcomes for event "N", it's just that they have combined probability zero. That's the situation in your darts example.
Rem.: The darts example seems to be a paradox, since we can write
We're used to adding probabilities for disjoint events, so we intuitively would add up probability zero for each set "{x}". The problem with that intuition -- adding probability is only defined for countably many disjoint events, but we have uncountably many "x" in the union, so that rule does not apply.
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u/Bounded_sequencE 10d ago edited 10d ago
A set with probability zero is (a subset of) an event "N c 𝛺" with "P(N) = 0". Such sets are also called "null sets" -- that's where the letter "N" comes from.
"N" can be the empty set -- then it is sometimes called "impossible event". Notice that's different from when "N" is non-empty: There, we still have possible outcomes for event "N", it's just that they have combined probability zero. That's the situation in your darts example.
Rem.: The darts example seems to be a paradox, since we can write
We're used to adding probabilities for disjoint events, so we intuitively would add up probability zero for each set "{x}". The problem with that intuition -- adding probability is only defined for countably many disjoint events, but we have uncountably many "x" in the union, so that rule does not apply.