Tabulate the probability distribution of $x$. > If a red dice and a green dice are rolled together and $X$ is the highest score minus the lowest score of the dice, what are the possible values of $X$? > > Tabulate the probability distribution of $x$. Personally, here is my solution based on my understanding: $$ P[X = 0] = \frac{1}{6} \quad\quad P[X = 1] = \frac{1}{6} \quad\quad P[X = 2] = \frac{1}{6} \\\ P[X = 3] = \frac{1}{6} \quad\quad P[X = 4] = \frac{1}{6} \quad\quad P[X = 5] = \frac{1}{6} $$ I know I answer wrongly. Who can explain and solve this problem?

See to answer this one you need to consider $(p,q)$ and $(q,p)$ as different results but (p,p) are considered as same results

Then we need to think how many possibilities are there if we throw red and green die simultaneously.There are 36 possibilities as follows:

$(1,1),(1,2),(1,3)....(1,6) $

$(2,1)....(2,6)$

...

$(6,1)....(6,6)$

then for getting $p[X=0]$ we have (1,1),(2,2)...(6,6) ,so there are 6 possibilites so $P[X=0]=\frac{6}{36}$ now doing similarly for $P[x=1]$ we have $(1,2),(2,1),(2,3),(3,2)....(5,6),(6,5)$ so we have 10 such possibilites. so $p[X=1]=\frac{10}{36}$

Now doing similarly for X=2,3,4,5,6 we get

$P[X=2]=\frac{8}{36}$, $P[X=3]=\frac{6}{36}$, $P[X=4]=\frac{4}{36}$, $P[X=5]=\frac{2}{36}$, $P[X=6]=\frac{0}{36}$