Probability of Maximum Value

Question

Let $X$ and $Y$ be independent, each uniformly distributed on $\{1, 2, ..., n\}$. Find $P(\max(X, Y) = k)$ for $1 \le k \le n$.

$P(\max(X, Y) = k) $

$= P(X = k \cap X > Y) + P(Y = k \cap Y > X)$

$= P(X=k)P(X>Y|X=k) + P(Y=k)P(Y>X|Y=k)$

$= \frac{1}{n} \cdot \frac{k-1}{n} + \frac{1}{n} \cdot \frac{k-1}{n}$

$= \frac{2(k-1)}{n^2}$

Textbook Answer: $\,\,\, \frac{2k-1}{n^2}$

Answer 1

$P(\max(X, Y) = k)$

$= P(X = k \cap X > Y) + P(Y = k \cap Y > X) + P(X = Y = k)$

$= P(X=k)P(X>Y|X=k) + P(Y=k)P(Y>X|Y=k) + P(X = Y = k)$

$= \frac{1}{n} \cdot \frac{k-1}{n} + \frac{1}{n} \cdot \frac{k-1}{n} + \frac1{n^2}$

$= \frac{2k-1}{n^2}$