Threshold Exceedance for the Minimum of Two Exponentials

Let $X \sim \operatorname{Exp}(2)$ and $Y \sim \operatorname{Exp}(3)$ be independent. Define $M = \min(X, Y)$ and fix a threshold $c = 1$.

(i) Find $P(M > 1)$.

(ii) Given $M > 1$, find $E[M - 1 \mid M > 1]$ and $P(X < Y \mid M > 1)$ (the probability that $X$ is the minimum, given both survived past 1).