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If the statement is true, prove it. If it is false, give a counterexample.

If $A$ is an event with probability 1, then $A$ is the sample space.

I know the statement is false. I just can't think of a counterexample.

So far, what I've come up with is the toss of a coin. Let $A$ be the event of a fair coin tossed. Let $E$ be the event of $A$ in which the coin toss results in an even number and let $F$ be the event in which the coin toss results in an odd number.

Therefore $P(A) = P(E) + P(F)=1 $

As the events of $E$ and $F$ are both $\frac{1}{2}$ then,

$\frac{1}{2} + \frac{1}{2} =1$

But this example covers the entire sample space of $A$.

Is there a better way to express this example? Please advise.

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  • $\begingroup$ Hint: look for continuous examples. Or you could have a discrete system with some points of probability $0$, but that seems artificial. $\endgroup$ – lulu Feb 15 at 21:51
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Consider the interval $\Omega=[0,1]$ with Lebesgue measure (length). Let $A=[0,1]\setminus \{p\}$ where $p\in[0,1]$.

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