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The game goes like this:

There are 20 cards to begin with, among which, 12 cards are marked with $1, 3 cards are marked with +2 picks, 2 cards are marked with +3 picks, 2 cards are marked with x2, and the last card is marked with x3.

  So, it is: $1,$1,$1,$1,$1,$1,$1,$1,$1,$1,$1,$1,+2,+2,+2,+3,+3,x2,x2,x3 

A player is allowed to randomly pick 5 cards from the shuffled deck. And if he picks the card such as +2, he is allowed to pick an extra 2 cards from the deck without replacement. And if he gets +2 or +3 again, he picks again, over and over. The overall prize a player can get is the product of cash prize sum and multiplier product.

  e.g.  a player might get this:  $1,$1,$1,$1,+2,x2,x2 So he gets $4 x2 x2 = $16 (multipliers applied multiplicatively) 

SO, what's the expected prize he can get?

This game cannot be modeled as a time homogeneous markov chain. Is there any model or any idea to solve this kind of problem rather than do a recursive dynamic programming on a pc?

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    After reading the post below I have one question: does the multiplier cards apply to the pot or to the total? What I mean by this is: Is the drawing (x2 $1 $1 $1 $1 ) equal to ($1 $1 $1 $1 x2)? Notice that if the 'times two' card is applied to the pot, the first drawing would be a total of 4$, while the second one would be $8. Because if they are indeed equal then his final statement is correct. If they are not, the problem is even harder.2012-06-15
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    @Damieh Multiplier always apply to the final total. Btw, do u know how to make comments friendly for dollar sign?2012-06-15
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    No I don't =(. How do you do it? Also, if the multiplier cards apply to the total and not the current stack, upvote bgins's answer! It's pretty neat! =)2012-06-15

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