[SumZero]

Given you don't care about order of the 7 cards the number of ways to get any 7 card hand is 52 choose 7. The number of ways to get 4 kings is 4 choose 4 * 48 choose 3. That is:

(1 * (48!/(45!*3!)))/(52!/(45!*7!))
=(48*47*46/(3*2))/(52*51*50*49*48*47*46/(7*6*5*4*3*2))
=(7*6*5*4)/(52*51*50*49)
=1/(13*17*5*7)
=1/7735

So if you round poorly once every 7735 trials might seem like once every 10,000 trials.

You are right that this is the same as dealing 45 cards with no king and if you look at your fractions you get:

48/52 * 47/51 * 46/50 * 45/49 * 44/48 * ... * 5/9 * 4/8

and if you note the terms cancel so you are left with:

(7*6*5*4)/(52*51*50*49)

which is the same as above.