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I never was happy with this even when I learned it. Which would explain why I'm having such a hard time with it now.

Vart, you don't ask really about P(A|B). You ask simply for P(A):

Note the "before" in your sentence. My draw happened before your draw, thus it is P(A), not P(A|B).
P(A|B), the probability that...

Still 1/13th. You drew the second card from the reduced pool of 63 cards with possibly only 3 kings. The math is ugly, but I think it would still come out right.

"Chance" is indeed a better word.

The strategy defines your winning probability, but your initial choice defines whether you actually win or lose a round.

Oh, great, now I can think of a good, readable explanation for the Monty Hall problem.

The initial choice is not between three doors. The choice is between two categories: good (33% chance) and...

You're still using intuition to counter counting. That won't work. The entire purpose of the puzzle is that it goes against intuition.

I hate statistics, but I love probability.

I will switch doors. After one of the wrong doors has been removed from play, the chance that I get the right door by switching equals the chance of hitting a wrong door on the first try, i.e. 66%.

Good quote indeed, but it's "lottery".

"Dice have no memory."

You can try to explain it to your friend this way: if I flip a coin 10 times, probability says that the most likely outcome is 5...