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I'm still confused about impossible events and events with zero probability.

  1. When the sample space is discrete and finite, does $P(A)=0$ ⇒ $A=∅$?

  2. Also, if B is an event which doesn't contain any elements from the sample space, is it therefore an impossible event $B=∅$? Like, rolling a $7$ on a $6$-sided die.

Thanks.

FishDrowned
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John Smith
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  • No, consider $\Omega = {1,2}, \mathcal F = 2^\Omega, P = \delta_{{1}}$. You are conflating sample space with other things. It is best to ignore the sample space and leave it as an abstract space. – Andrew Nov 19 '23 at 05:51
  • Is "impossible event" a technical term in probability theory? How is it defined? – user14111 Nov 19 '23 at 07:03
  • An impossible event is an event that is not part of a sample space. i.e. $\forall \mathcal{F} \cap \Omega = \emptyset$ – FishDrowned Nov 19 '23 at 07:11
  • You cannot consider events outside of sample space. That doesn't make sense. Probability is always relative to some measurable space. To consider rolling $7$ on a $6$-sided die you need a model (probability space) that includes such event. The probability can be $0$, but formally needs to be included. – freakish Nov 19 '23 at 07:44

1 Answers1

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  1. No. Let $\Omega=\{a,b\}$, and $P(a)=1$, $P(b)=0$. Then the event $B=\{b\}$ is not empty but has probability zero.

  2. By definition, ``event'' is a subset of $\Omega$. If $B$ does not contain elements of $\Omega$ and is still a subset of $\Omega$, it means $B=\emptyset$.

van der Wolf
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