# If A is an event with probability 1, then A is the sample space: true or false

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.

• Hint: look for continuous examples. Or you could have a discrete system with some points of probability $0$, but that seems artificial. – lulu Feb 15 at 21:51

Consider the interval $$\Omega=[0,1]$$ with Lebesgue measure (length). Let $$A=[0,1]\setminus \{p\}$$ where $$p\in[0,1]$$.