Relation between expectation and sample points

Question

Suppose that the expectation of a random variable $X$ is $5$. Which of the following statements is true?

1. There is a sample point at which $X$ has the value $5$.
2. There is a sample point at which $X$ has value greater than $5$.
3. There is a sample point at which $X$ has a value greater than equal to $5$.
4. None of the above

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My attempt:

One of the friends, explained it as:

$E(x)=p_1x_1+p_2x_2+......+p_nx_n=5$

$p_1+p_2+.....+p_n=1$

And, $0\leq p_i \leq 1$

So, $x_i \geq 5$

But, why option $(1)$ and $(2)$ are not true?

Can you explain in formal way? Please.

Answer 1

The first is certainly not correct. Your sample points could be only $10,0$ or something like that, with average $5$.

The second one, you could have all sample points as $5$.

The third one has to be correct, because if all sample points are strictly smaller than $5$, their average is strictly smaller than their supremum, which is $5$. Thus, the third one is right.

Answer 2

1 is not true : For instance you have an average of 5 with 2 samples point 4 and 6. 2 is not true as you can have all your samples point equal to 5. in that case, your average would be 5 but you won't have any point above 5.