A Curious Adventure of Chances

Discrete Mathematics Level pending

I was telling the following probability adventure to my sister.

I and my friend Prash were staring at a finite sample space \( \omega\), when the following conversation took place:

I: I am thinking of two independent events \(A\) and \(B\) in \( \omega\).

Prash: Are both events non trivial?

I: You mean the probabilities of the events is strictly between \( 0 \) and \(1\) for both, right? Yeah, it is.

Prash: (evil smile) I know the probabilities of both events!

I: What?! I didn't tell you anything about my events!!

At this point she asked me, "Did Prash know the sample space? ".

I agreed that he did. She then remarked, "Even I can tell the probabilities of both events". As she left the room, she further pointed out "I can even tell the number of elements in the sample space!".

I was surprised. She didn't even know the sample space and yet she knew what Prash knew.

Can you tell the number of elements in the sample space?

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