[aces961]

oops, I realized I didn't type all I meant to before, I need to sleep more, I'll clarify the notation as well.
that sentence should read.
"When you make the statement if he picks 3 out of the hat there is a 50 percent chance that N<6. This is stating that SUM over i of P(3 given N=i)*P(N=i) from i =1 to 6 is equal to

SUM over i of P(3 given N=i)*P(N=i) from i =7 to infinity"


P(3 given N=i) is the probability 3 is chosen given N=i. In your example this is 1/N for n 3 or greater.

These sums being are equal are completely dependent on what the distrubution of P(N=i) is for each i. Also I changed this to <=6 but you can easily change the sums to 1 to 5 and 6 to infinity it doesn't change any of my arguement.