Question

Suppose we roll two dice and let X be the maximum of the two numbers obtained and let Y be their minimum. Write down the distribution tables of X, Y and Z = (X, Y ).

Answer 1

Number i is max number on two dice means that on one dice rolled i and on the other one rolled i or less, so, for every i there will be 2*i - 1 such possibilities(for every pair except (i,i) exists symmetric one)

Total possible outputs when rolling two dice is $6^2=36$. So,

$P(X=i)={\frac {2i-1} {36}}$

Number j is min number on two dice means that on one dice rolled i and on the other one rolled j or greater, so, for every j there will be 2*(7-j) - 1 such possibilities(for every pair except (j,j) exists symmetric one)

Total possible outputs when rolling two dice is $6^2=36$. So,

$P(Y=j)={\frac {2(7-j)-1} {36}}$

$\forall (i, j\in [1;6])\cap (i<j):P(X=i, Y=j)=0$ - cause the max value cannot be less than min value

$\forall (i, j\in [1;6])\cap (i>j):P(X=i, Y=j)=\frac 2 {6^2} = \frac 2 {36}$ - cause for every (i,j) there is symmetric one

$\forall (i, j\in [1;6])\cap (i=j):P(X=i, Y=i)=\frac 1 {6^2} = \frac 1 {36}$ - cause for every (i,i) there is no symmetric pair