Expected lifetime Exponential random variable

Question

The lifetime of a dog and a cat are independent exponential random variables with rates $\lambda$ and $\mu$ respectively. One of them just died. ¿What is the expected value of the lifetime of the pet that just died?

I don't know which one would be correct

Let $L ={}$lifetime of the pet that died

$L_d ={}$lifetime of dog ($L_d \sim \operatorname{Exp}(\lambda)$)

$L_c ={}$lifetime of cat ($L_c \sim \operatorname{Exp}(\mu)$)

1. $E[L]= E[\min(L_{d},L_{c})] = \dfrac{1}{\lambda+\mu}$

or

2. $E[L]= E[L\mid L_d>L_c]P(L_d>L_c) + E[L\mid L_c>L_d]P(L_c>L_d) \\ = E[L_c]\dfrac{\mu}{\mu + \lambda} + E[L_d]\dfrac{\lambda}{\lambda + \mu} = \dfrac{2}{\lambda + \mu}$