exponential and gamma distribution, survival functions

I have a homework problem, that I believe I can solve correctly, using the exponential distribution survival function. But, I think, I should also be able to solve it more easily using a gamma distribution formula? Could you please help? Thank you.

The magnitude of earthquakes recorded in a region of North America can be modeled as having an exponential distrubiton with mean 2.4, as measured o the Richter scale. Of the next ten earthquakes to strike the region, what is the probability that at least one will exceed 5.0 on the Richter scale?

So, if I model this using an exponential function, we have:

$r.v. X\sim\exp(\beta=2.4)$

Survival function, for one trial:

$P(X>5) = e^{-5/2.4} \approx 0.1245$

So, I think the answer is just:

$1 - (1 - e^{-5/2.4})^{10}$

Right? But, I think there should be some easier way to solve it using a gamma distribution survival function? I'm getting confused when I try it.

$r.v. X\sim gamma(\alpha = 10, \beta = 2.4)$

It was my impression that the survival function is as follows:

$f_x(X) = \dfrac{X^{\alpha-1}e^{-X/\beta}}{\Gamma({\alpha})\beta^{\alpha}}$

But, I must be applying it wrong, because I get:

$f_x(5) = \dfrac{5^9e^{-5/2.4}}{\Gamma({10})2.4^{10}}$

This doesn't evaluate to the same probability as my method using the exponential distribution. Could you please help me determine where I'm going wrong? Thank you!

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Under the assumptions, including the unstated (and wrong: aftershocks) independence, the first calculation is correct, and surely simple. – André Nicolas Apr 9 '13 at 5:55
But, isn't there a way to use gamma distribution? Am I applying the gamma distribution incorrectly? I must be... – Clark Henry Apr 9 '13 at 5:58