Largest and smallest values of 2 dice

Question

Two dice are rolled. Let $X$ and $Y$ denote, respectively, the largest and smallest values obtained. Compute the conditional mass function of $Y$ given $X = x$, for $x = 1, 2, · · · , 6$. Are $X$ and $Y$ independent?

The answer to the first question about the conditional pdf is given by the following table.

Now, I can understand how to manually get these values, for example, the value of $X=2$ and $Y=2$ being $1\over3$ from the fact that there are three values that involve the greatest number being $2$, which are $(1,2) , (2,1) , (2,2)$.

However, I am still confused about how to use the formula $P(Y|X) = \frac{P\{ X=x, Y=y\}}{P\{X=x\}}$ to get these values. What is the $P\{ X=x, Y=y\}$? What is the $P\{X=x\}$?

Answer 1

$P\{ X=x, Y=y\}$ is always either $1\over{36}$ if the numbers are the same, or $1\over{18}$ if they are different, or $0$ if $Y>X$.

$P\{X=x\}$ is always the number of times out of $36$ combinations that a particular number is the greatest, in my example of $X=2$ and $Y=2$, it is $3\over{36}$ as there are three choices $(1,2) , (2,1) , (2,2)$ out of $36$.

Answer 2

Let's consider the third entry (under the column heading $3$) in the first row of the table (the row labeled $p_{Y\mid X}(1\mid x)$). For this entry, $y = 1$ and $x = 3,$ so $$P\{ X=x, Y=y\} = P\{X=3, Y=1\},$$ that is, it is the probability of rolling a $1$ and a $3$ (in either order), and $$P\{X=x\} = P\{X=3\},$$ that is, it is the probability that at least one die shows a $3$ and the other does not show a higher number.

Answer 3

If the largest number is $x$ the smaller number is either equal to $x$ (in one way) or smaller than $x$ (in $2(x-1)$ ways), so $$Pr(X=x)={2x-1\over36}$$

As for $Pr(X=x, Y=y)$ this can happen in no ways if $y>x,$ in one way if $y=x,$ and in two ways if $y<x,$ so $$Pr(X=x,Y=y)= \cases{0,&$y>x$\\{1\over36},&$y=x$\\{1\over18},&$y<x$}$$