Given $\displaystyle X_1,X_2,X_3$ are independent and exponentially distributed random variables with respective rates $\displaystyle \lambda_1,\lambda_2, \lambda_3$.

How can I derive $\displaystyle P(X_1=min(X_1,X_2,X_3))$.

In fact I know the answer, it should be $\displaystyle \frac{\lambda_1}{\lambda_1+\lambda_2 + \lambda_3}$.

Meanwhile I also know that $\displaystyle P(X_1<X_2)=\frac{\lambda_1}{\lambda_1+\lambda_2}$

But I just can't seem to get it :(