Best of Two Javelin Throws

Alice and Bob each throw a javelin twice, and a competitor's score is the longer of their two throws. Every Alice throw is Exponential(rate 2); every Bob throw is Exponential(rate 1); all four throws are independent. Let $X$ be Alice's score and $Y$ be Bob's.

Compute $P(Y > X)$.

Answer format: round to 3 decimal places.