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So this is an example straight from a book, like an example to help teach the material yet it makes absolutely no sense or I am just not seeing where they make the jump at.

We have the following: Consider the experiment consisting of 2 rolls of a fair 4-sided die. Let X be a random variable, equal to the maximum of the 2 rolls. It then says complete the following table (I don't know how to make a table on here)

| $\space$$\space$$\space$$\space$x$\space$$\space$$\space$$\space\,\,$ | 1 | 2 | 3 | 4 |
|Pr(X=x)| ? | ? | ? | ? |

With the sample space S ={(1, 1),(1, 2),(1, 3),(1, 4),(2, 1),(2, 2),(2, 3),(2, 4),(3, 1),(3, 2),(3, 3), (3, 4),(4, 1),(4, 2),(4, 3),(4, 4)}

I don't see how Pr(X=1) = $1\over16$, Pr(X=2) = $3\over16$, Pr(X=3) = $5\over16$ and Pr(X=4) = $7\over16$.

I can see that there is a total of seven 4's in the sample space but there's also seven 1's in the sample space, same with seven 2's and so on. Could anyone give a bit of a hint on what I'm missing? I've read the definitions and everything leading up to this example but they don't help.

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    The sample space contains tuples, not numbers. What do you mean that it contains $4$'s and $1$'s? – 2017-02-24
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    There are 7 pairs with a 4 in them somewhere. 7 pairs with a 1 in there somewhere and so on is what I mean. It's the only thing I can see that gives me a 7. – 2017-02-24
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    $X$ is the *maximum* of the two rolls. So $\operatorname{Pr}(X = x)$ means: how many rolls of two dice give two numbers of which $1$ is the maximum? In the case of $x = 1$, for example, it's easy to see that the only way to have $1$ as maximum is to roll $(1, 1)$. This is just one case out of the $16$ totals, giving $1/16$ as a result. – 2017-02-24
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    @rubik I finally realized what I missed. I thought "Let X be a random variable, equal to the maximum of the 2 rolls" meant that there was only two rolls total... – 2017-02-24
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    Perhaps what you wrote makes sense to you in some strange way but there *are* only two rolls total... you take a 4-sided die roll it, record it, roll it again, record it (*or alternatively take two dice and roll them both*), and look to see what the larger of the two recorded numbers are. Whatever the larger number of the two rolls are is going to be the value of $X$ that we witness in that specific instance of the experiment. – 2017-02-24

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For example, $X=3$ consists of the sample points $(1,3), (2,3), (3,1), (3,2), (3,3)$. There are $5$ of them, while the sample space has $16$ points, all equally probable, so $P(X=3) = 5/16$.

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    Wait, does this mean X=3 is saying that the largest number in the pair is 3? – 2017-02-24
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    @Heavenly96 that is what maximum means – 2017-02-24
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    @user251257 I thought "Let X be a random variable, equal to the maximum of the 2 rolls" meant that there was only two rolls total. Thank you for the clarification, I misread the question very badly. – 2017-02-24