Probability that the millionth decay occurs within 100.2 seconds? 
Radioactive decay of an element occurs according to a Poisson process
  with rate $10,000$ per second. What is the approximate probability
  that the millionth decay occurs within $100.2$ seconds?

Let $X$ be the number of decays and the number of expected decays within $100.2$ seconds is $\lambda=100.2\cdot10000=1002000.$ Thus $\bar{X}\sim \text{Poi}(1002000)$ and $\mu=1002000, \ \sigma=\sqrt{1002000}=1000.995.$
How to I formulate "probability that the millionth decay occurs within $100.2$ seconds?" 
Is it $P(X>1000000)?$ I don't se how.

You have a Poisson process with expected value $1,002,000$.  The chance that the millionth decay has not happened yet is the sum of the probabilities of exactly $0,1,2,\ldots ,999,999$ decays having happened.  I believe you are supposed to use the normal approximation.  Based on the figures you quote, a million decays is just about $2\sigma $ low so you need the chance that a random normal is greater than mean-$2\sigma $.
