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If I have a standard deck of 52 cards with ace being 1 and jack being 11, queen 12, and king 13. I know the expected value of any card I draw to be 7. However, how does this change with each subsequent draw?

For example, if I draw a king on the first turn. Then is $E(\text{next card}) = \frac{48}{51} \left( \frac{13}{2} \right) + \frac{3}{51} \left( 13 \right) = 6.88$. How can this be generalized to $E(\text{next card} | x_i \text{ is drawn})$ or $E(\text{next card} | x_1,x_2,..,x_n \text{ are drawn})$

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Since

$ \sum_{i = 1}^{13} 4i = \frac{4(13)(14)}{2} = 364 $

Note 364÷52 = 7. That's how we get the expected value of a card drawn from a full deck.

You can simply count the total number of points in your hand , subtract that total from 364 and divide by how many cards are left in the deck to get the expected value of the next card drawn.

$ E = \frac{364 - p}{52 - c} $

Where p is the sum total of all card values already drawn and c is the number of cards already drawn.

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