d) compute $\displaystyle P\{X_1=\min\{X_1, \ X_2\}\}$ with rates $\displaystyle \lambda_1$ and $\displaystyle \lambda_2$

based on my book the probabilistic minimum of is just $\displaystyle \frac{1}{\sum_{i=1}^n \lambda_i}$ (no proof was given) so in this case it would be $\displaystyle \frac{1}{\lambda_1+\lambda_2}$ also I would like to show that $\displaystyle \frac{1}{\sum_{i=1}^n \lambda_i}$ is the case, with 2 variable. I thought it would be:

$\displaystyle P\{X_1, < X_2\} + P\{X_2, < X_1\}$ but that give me 1 and

another way I considered was $\displaystyle P\{X_1, < X_2\} \cdot P\{X_2, < X_1\} = \frac{\lambda_1 \lambda_2}{(\lambda_1 +\lambda_2)^2}$

I finally considered $\displaystyle n[1-F(y)]^{n-1} \cdot f(y)$ but I don't know what $\displaystyle \lambda$ to pick since they are both different.