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Let $A$ and $B$ be two events in the sample space $S$. Describe & represent the following event in terms of $A$ and $B$: "at most $A$ does not occur".

This is a question that someone asked me, and I think it is meaningless as "at most" makes no sense here.
Am I right? Or it means that $B$ must occur and $A$ mustn't occur.

Please help and share your thoughts and ideas if possible. Thanks.

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  • In general, "at most" means "not more than": in this case this means that "both A and B do not occur" is not possible. Thus, the case "A and B occur" is possible. – Mauro ALLEGRANZA May 08 '24 at 07:06

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It is "Poor Wording" , I concur that it is almost meaningless.

Here are 3 ways to look at it :

INTERPRETATION 1 :
In general : $A$ might not occur , $B$ might not occur.
We want to consider the Cases where at most $A$ does not occur.
We do not want the Case where $B$ too does not occur.
Hence we want the Cases where $B$ occurs , though $A$ might or might not occur.

INTERPRETATION 2 :
In general : "at most $A$ might not occur" is "at least $A$ might occur".
We want to consider the Case where $A$ does occur , whether or not $B$ occurs.
Hence we want the Cases where $A$ occurs , though $B$ might or might not occur.

INTERPRETATION 3 :
In general : $A$ might occur , $A^C$ might occur , $B$ might occur , $B^C$ might occur , $A \cup B$ might occur , $A^C \cup B$ might occur , $A^C \cap B$ might occur . . . .
We want to consider the Cases where at most $A$ does not occur.
It is then not clear whether we want to consider $A^C \cup B$ , $A^C \cap B$
We do not know what the "at most" is supposed to restrict here , where we have no linear order on the Events , unlike "at most 6 heads" versus "at most 8 heads" where Natural Numbers have a common universal linear order.

We can easily see that all 3 ways are not matching.

It is "Poor Wording" : though the writer or speaker might have some unconscious meaning attached to that term , it is not universal.

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