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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?

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Your mistake is in multiplying the two events' probabilities to get the probability that both occur. You can only do this if the two events are independent, which they are not in this case.

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

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