I'm trying to solve the following question:
We roll a pair of dice. Let $X$ be the random variable which represents the first result, while $M$ represents the highest result out of the two. Find the joint probability function of $(X, M)$.
The solution:
For every $(k,m)$ we have $k\in\{1,2,3,4,5,6\}$, $m\in \{k,k+1,\ldots,6\}$. So we get - for $1\leq k<m\leq 6$, $P(X=k,M=m)=\frac{1}{36}$ and for $1\leq k=m\leq 6$, $P(X=k,M=m)=\frac{m}{36}$
I'm trying to figure out why. Clearly the random variables are dependent on each other, So I can't use: $$ P(X=k,M=m)=P(X=k)\cdot P(M=m) $$ But as I understand $P(X=k)=\frac{1}{6}$ so it should be $P(M=m)=\frac{1}{6}$.
Anyway, I'm trying to understand why they got $\frac{1}{36}$ and $\frac{m}{36}$. I'm trying to use the $P(A)=\frac{|A|}{|\Omega|}$ method here. Is it possible to explain how did they got it?
If $k<m$ there is only one way to get this combination: roll $k$ on the first die and $m$ on the second.
If $k= m$ there are $m$ ways to get that combination: roll $m$ on the first die and any number from $1$ to $m$ on the second.
Each "way" has probability $1/36$, since it involves rolling a specific number on each die.
As a sanity check, the total of all these probabilities is $15\times \frac{1}{36}+\frac{1+2+3+4+5+6}{36}=\frac{36}{36}$.