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Linear Approximation to the Infinite Deck Blackjack Function
  The phrase `infinite deck blackjack' can be interpreted in two ways: either as the limit of considering increasingly large finite decks or as the result of dealing with replacement so the chance of a card's appearance at any stage of the hand is not altered by whether it has or has not appeared earlier in the deal. The two interpretations coincide, since the limiting probabilities in the first case are the same as those occurring from the independent dealing of the second.
Adopting the first perspective, we deduce that if all possible subsets of some fixed and finite size (52 for instance) were selected from an infinite deck (independent sampling), the average expectation of these subsets, probabilized without replacement, would be the same as for the infinite deck. We shall see shortly that this average expectation when probabilizing with replacement is not the same, but rather has a consistent and predictable bias from the full infinite deck figure.

 


 
 
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