# What is wrong when derive the Minimum of Three Independent Exponential Random Variables in such a way?

Let $$X_i, i=1,2,3$$ be independent exponential random variables with rates $$\lambda_i, i=1,2,3$$.

I know the right answer is:

$$P\{X_1 = min(X_1, X_2, X_3)\} = \frac{\lambda_1}{\lambda_1 + \lambda_2 + \lambda_3}$$.

And someone has shown a detailed proof out there.

However, I get a different equation if I try

$$P\{X_1 = min(X_1, X_2, X_3)\} = P\{X_1 = min(X_1, X_2), X_1 = min(X_1, X_3)\} = \frac{\lambda_1}{\lambda_1 + \lambda_2}\frac{\lambda_1}{\lambda_1 + \lambda_3} \neq \frac{\lambda_1}{\lambda_1 + \lambda_2 + \lambda_3}$$.

What is my mistake?

(The two events here being $$\{ X_1 =\min(X_1, X_2)\}$$ and $$\{ X_1 = \min(X_1,X_3)\}$$.)