Probably of winning with 2 dice against 1

Question

could some of you help me to find out what is the probability of A) obtain with two dice a greather number than another die? B) and if the dice are 3 how can I do? Not the sum of the 2 dice, but the greatest value of those 2 against another die

Answer 1

If $X$ is the sum of the two dice, and $Y$ the value of the one die, then what are the possible outcomes $(x,y)$ and which one are "winning"? Then compute their probabilities by independence and add.

Or compute when the one die wins :

$P(Y=6)P(X \le 5) + P(Y=5)P(X \le 4) + P(Y=4)P(X \le 3) + P(Y=3)P(X \le 2)$

which equals $\frac{1}{6}(\frac{1}{36} + \frac{3}{36} + \frac{6}{36} + \frac{10}{36})$

and take the complement of that.