Dice rolling and independence

Question

Two dice are rolled. $X$ is the smallest and $Y$ is the largest value that can be obtained on rolling the dice. Are $X$ and $Y$ independent?

How to check that 2 joint probability mass function are independent? I had already read the text book and it said that X and Y are independent if $P\{X=i,Y=j\}$ equals to $P\{X=i\}P\{Y=j\}$

So from the question above, I had already tried this question and I found out that it’s not independent.

Did I get it right?

Answer 1

Yes, that all looks good to me, except that you have got $X$ and $Y$ the wrong way round. You don't need to compute the conditional probabilities to show independence (though that's certainly a valid way to do it). Instead, simply multiply any two of the marginal probabilities in the first table together, and check you don't get the corresponding entry in the table. For example $P(Y=6)=\frac{11}{36}$ and $P(X=6)=\frac{1}{36}$ (in the corrected table) but $P(X=Y=6)=\frac1{36}\neq\frac{11}{36}\times\frac{1}{36}$.

In fact it's even simpler: $P(X=4)>0$ and $P(Y=3)>0$ but $P(X=4,Y=3)=0$ so they can't be independent.

Answer 2

A superficial analysis,

if $X = 6$,then $Y$ must be $6.$ While if $X = 1, Y$ could be anywhere in $[1,6]$ Hence $X,Y$ cannot be independent.