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However, I still don't understand why due to symmetry the probability for getting black jack is the same for either you are the dealer. When the cards are dealt out, a total of 4 cards are dealt out. How come this doesn't come into play? Say one person is dealt first. Now you have 50 cards to deal the other player from. Wouldn't you calculate the probability of the second player getting blackjack based on the cards dealt to the first player?
Also, in a real blackjack game, the player gets one card and then the house. Doesn't that once again change the probability of getting blackjack for the two players?