I'm interested in calculating the chance of receiving certain cards given the player's hand and the dealer's up-card after the dealer has "peeked" for Blackjack.
Typically, if the dealer shows an Ace or 10-value card (10,J,Q,K) the dealer will "peek" at their second card and check for Blackjack. If Blackjack is discovered all players (except those with Blackjack also) will lose immediately and the next round is dealt - this speeds up the game and thus speeds up casino profits.
If the dealer does not peek for Blackjack this calculation is easy:
For example, suppose the dealer shows a 4 and the player shows 2,3. Then the probability our next card will be a 5 is simply 4/49 (4 5's left in 49 remaining cards). The dealer's down-card is equally likely to be any of the 49 remaining cards and thus shouldn't affect the probability at all, we simply ignore it.
However, if the dealer shows a 10 and the player shows 2,3 and the dealer doesn't have Blackjack things are different. Intuitively, because we know the dealer doesn't have an Ace, we should be more likely to receive an Ace (or less likely to receive a non-Ace) - but I'm unsure how to calculate this exactly.