Random Value

Question

Draw 5 cards at random from the deck. For sums < 8 you win double that sum in dollars. For 8 ≤ sum ≤ 20 you lose half your sum in dollars. If sum > 20, you lose total sum in dollars. Find the expected value of the game.

Answer 1

This one is really difficult.  The best approach I can come up with is to start by breaking down the probability that the hand dealt will contain 0, 1, 2, 3, 4 or 5 10-counts.   

 

Once you have that probability, you can further break down each of the 5 non-full hands to determine based on random pull of the other 1-5 cards what the probabilities are for each number of points to be added to the 10-count baseline.

 

This is very calculation-intensive, though, so I find it hard to imagine it being on a typical statistics course.  Is this an extra credit problem of some sort?