The sum of 2 poissons is obviously poisson just like the sum of 2 binomials is binomial,ur reasoning is flawed,if X ε Poi(λ) and Y=X+X,THEN Y ε Poi(2λ),it follows quite simply evn by general reasoning.
My suggestion is that you should try the moment generating function technique,ofcos u will need the theorem on limiting mgfs,bt it is the easiet way,or do a transformation
To me your explanation is simple enough so I think you should tell your friend that a head is also possible on the 1st toss,we are 100% sure of getting atleast 1 head,not where that head will be i.e. 1st or 2nd toss.
Mathman is right,getting a kazillion 9s is not equiprobable to getting an even distribution,i.e it can only happen in 1 way while the numbers can be evenly distributed a number of different ways e.g. the probability of 6000 6s in 6000 throws of a die is not even close to being equal to the...