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My lecturer is adamant that the following is true and I'm struggling to get my head around it:

"relation A ⊂ B between two events A and B means that the event A occurs whenever event B does, but not necessarily the opposite."

My understanding is that this notation means that A is a proper subset of B. It makes sense to me that the opposite would be true - that event B occurs whenever event A does, but not necessarily the opposite, because there are elements of B that are not in A. Can someone please explain why I'm wrong?

Andrés E. Caicedo
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YS731
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    I think it is you who is correct. – Swapnil Rustagi Nov 22 '16 at 17:57
  • Your reasoning is not in sync with your conclusion. – StubbornAtom Nov 22 '16 at 18:06
  • @StubbornAtom please can you explain why – YS731 Nov 22 '16 at 18:12
  • @YS731 $A$ is a subset of $B$. This means that $A$ is contained in the bigger set $B$. So whenever $B$ occurs, $A$ necessarily occurs as every element of $A$ is in $B$. But there may be elements of $B$ not in $A$ ( as $B$ is the bigger set), so occurrence of $A$ does not necessarily imply occurrence of $B$. – StubbornAtom Nov 22 '16 at 18:26
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    @StubbornAtom I think maybe we differ on what it means for an event to occur... My understanding is that an event occurs whenever any of its elements are an outcome of the experiment. It doesn't have to be all of its elements. So if a single element of A is an outcome, A has occurred. That same element is an element of B, because A is a proper subset of B, so B has also occurred. Like you said, there are elements of B not in A that can be an outcome, so B occurring doesn't mean A has occurred. – YS731 Nov 22 '16 at 19:05
  • @YS731 That is indeed how events are defined in probability work. – Graham Kemp Nov 23 '16 at 00:20
  • Are you sure that your lecturer did use $\subset$ as (set) inclusion ? – Mauro ALLEGRANZA Nov 23 '16 at 06:53
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    @MauroALLEGRANZA he used that symbol and then showed a venn diagram of a small circle (A) inside a big circle (B). Could there be another meaning for the symbol? I did wonder if maybe he meant implication – YS731 Nov 23 '16 at 18:53

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Suppose your outcomes are: (r)ain, (s)shine), (d)rizzle, (h)ail. So your sample space is $\Omega = \{r, s, d, h\}$.

Now, let $A$ be the event "rain or shine": $A = \{r, s\}$, and let $B$ be the event "rain or shine or hail": $B = \{r, s, h\}$.

If the weather is "hail" (outcome $h$), then event $B$ has occurred, but event $A$ has not.

However, if $A$ occurs, then so necessarily does $B$.

avs
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