Dan, Dominic, and Doug
are waiting together in the living room for their girlfriends, Sally, Shellie, and
Susanne, to call. Their waiting times (in hours) are independent Exponential
random variables, with parameters $2.1$, $3.7$, and $5.5$, respectively. What is the
probability that the phone will ring (i.e., the first call will arrive) within the
next $30$ minutes (i.e., within the next $1/2$ an hour)?

so $\lambda\_1 = 1/126$ for Dan

$\lambda\_2 = 1/222$ for Dominic

$\lambda\_3 = 1/330$ for Doug

Let X represent Dan's waiting time. Y represent Dominic's waiting time. Z represent Doug's waiting time.

$P(X\leq30) = 1 -e^{-30/126}$

$P(Y\leq30) = 1 -e^{-30/222}$

$P(Z\leq30) = 1 -e^{-30/330}$

so $P(min (X,Y,Z) \leq 30) = ???$

I don't know what to do in this last step. The final answer is $0.9965$ But how did they get that answer?