I will post the Question and the specific part of the Mark scheme I have an issue with verbatim below to ensure my question is understood, I don't understand the logic behind a specific part of the mark schemes answer.
In the game of Risk, rolling six-sided dice decide the outcome of battles. Suppose that there are two players, red and blue. The red player rolls three dice $X_1, X_2, X_3$; and the blue player rolls two dice $Y_1, Y_2$. Whichever player rolls the highest number on a single die is declared the winner. In the event of a tie, the blue player is declared the winner. Assume independence between dice rolls.
(a). Let $X$ and $Y$ be the highest number rolled by the red and blue player. What is the probability mass function of $X$? What is the pmf of $Y$?
(b). What is the probability that the red player wins?
So, my issue is with the answer to logic to the answer in part $(b)$ where it says: The probability that the red player wins is $$P(Y < X) = P(Y ≤ X − 1)$$ Why is it that subtracting the range of values of the Random variable $X$ results in $Y$ being less than or equal to $X-1$, what about when $Y = 3$ and $X = 5$, would $X$ not still win and thus should be included in the calculation of the probability.
I believe my question is: shouldn't the probability include the cases where $Y \leq X - n$ where $n = \{1,2,3,4,5\}$ to account for all cases where $X$ is greater than $Y$?
A proof is also included which may help but I don't fully understand it so perhaps an explanation of the logic behind the proof will suffice in answering my question.
