This is a complicated question but I'll simplify the answer.
If you are talking about getting 21 on the inital deal of two cards then that is a fairly easy calculation. You need to first find the chances of getting a 10 card and then find the chances of getting an 11 card.
10 card
out of 52 possible cards there are 4 each of: 10s, Js, Qs, Ks which means you have 16 ways of getting 10 out of 52 cards.
11 card
out of 52 possible cards there are only 4 kinds of Ace so 4 out of 52.
Then you multiply the probability of getting each card and you have your answer: (16/52) * (4/52) = 64/2704 = 2.37%
The answer can get more complicated when you start to think about how there is only 51 cards left after you take the first card etc.
It is true that you have 16 cards representing "10" and 4 representing "11" in a single deck of cards.
To calculate the probability to get "
Black Jack" or 21 on the initial deal, we need to consider both events
A: first card "10" and second "11"
B: first card "11" and second "10"
We have
\(\displaystyle P("Black Jack") = P(A) + P(B) =\frac{16}{52}\cdot\frac{4}{51}+\frac{4}{52}\cdot\frac{16}{51}=\frac{32}{663}\approx 0.04827\)
where we take into account the fact that only 51 cards remain in the deck after first card is drawn. This is the probability of getting blackjack when using one deck of cards and the deck is "
perfectly" shuffled between subsequent deals.
It corresponds fairly to 1/20.7 as proposed by "Playing Blackjack as a Business" accordning to kahlmy_ishmael_xxiii's post.
Also, don't forget about the fact that most casinos use multiple decks which makes the odds even worse for you.
Suppose we have n decks perfectly shuffled between subsequent deals in a one player Black Jack game. Let the events be A and B as denoted above.
Now
\(\displaystyle P(A)=P(B)=\frac{16n}{52n}\cdot\frac{4n}{(52n-1)}=\frac{16}{13}\cdot\frac{n}{(52n-1)}\)
\(\displaystyle P("Black Jack") = P(A) + P(B) =2P(A)=\frac{32}{13}\cdot\frac{n}{(52n-1)} \)
With n = 4 we get a probability of 128/2691 or about 0.04757. This is close to the proposed value of 1/21 by "Playing Blackjack as a Business" accordning to kahlmy_ishmael_xxiii's post.
The situations get more complicated as many hands are dealt after another without shuffling. Also... to find the probability to get 21 with many cards, knowing what cards are dealt, maybe knowing the number of decks, maybe knowing the decks aren't tampered with etc. ...is quite a complicated task...