Probability of Normal Distribution

Question

Let's say that 10 sumo wrestlers were to squeeze into an elevator that could only hold a max capacity of 2300 pounds. Let's say that the weight of the sumo wrestlers is normally distributed with a mean of 220 lbs and a standard deviation of 25 pounds, what's the probability that the elevator is past it's max cap.?

So far I have that

μ = 220 σ = 25

I thought of (2300-2200)/25 = 4 (but that doesn't seem right. )

and something like P {X > 2300} = Pr {Z>what apparently 4 would be} = 1 - (something here)

I'm not sure if I'm even doing this problem right.

Answer 1

Let $X_i=\text{N}(220,25^2)$ be the weight of the $i^\text{th}$ wrestler.

Then $X=\sum_iX_i$ be the total weight.

Now read http://en.wikipedia.org/wiki/Sum_of_normally_distributed_random_variables and http://en.wikipedia.org/wiki/Fubini's_theorem and note that Fubini's theorem applies to random variables.