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Draw!
Applied Problems - Draw Poker

 

 In the game of draw poker, the idea is to get the best 5 card hand possible.

 The players bet that they have a better 5 card hand
 than any other player remaining in the game.
 If you don't think you have the best hand, 
 and don't think that you can trick the other players
 into thinking that you do (called bluffing), 
 you can quit for that hand (called folding).
 The rules of what beats what are a bit complex,
 but basically more cards of the same number 
 or more cards of the same suit
 or having all cards in numerical order, is a good thing.
 
 Poker is more a game of human nature and Psychology,
 but it helps to know the odds.
 Leaving out the betting, 
 the way the game is played is that each player is dealt 5 cards.
 The player can keep all of the cards or trade in some in
 for replacement cards from those left in the deck. 
 You don't get to choose what replacement cards you get. 
 You basically get them at random.
 
 Say you are in the game and get these 5 cards ...
 
10 of Hearts
10 of Spades
6 of Hearts
Jack of Hearts
Queen of Hearts
 
 You now have to decide if you want to keep these cards
 or trade some of them in for cards that might be better.
 
 Here are some strategies you might consider ...
 

  1) Keep the four hearts. Trade in the ten of spades

  and hope you get another heart so you have cards that are all one suit.
 
  2) Keep the two tens. Trade in the other cards
  to try to get more tens.
 
  3) Keep the ten, Jack, and Queen of Hearts.
  Trade in the 6 of Hearts and the10 of Spades and try to get a Royal Flush.
  That is, a ten, Jack, Queen, King, and Ace all in the same suit.
 
 What are the chances of each of these options working?
 
 There are 52 cards in a deck. You have 5 of them.
 That means there are 47 cards from the deck that you don't have.
 

  Other players have some of these cards,

  but you don't know which ones they have.
 That means we treat all other cards the same way.
 
 Say you try for the flush (choice 1)
 You trade in the 10 of Spades.
 You hope to get another Heart card.
 There are 13 Hearts in the deck.

 You have 4 of them. 

 That means there are 9 other Heart cards in the other 47 cards in the deck.
 The chance of getting one of them is ...
 

 

 OK, Now let's try for more tens (choice 2)

 What is the chance that you will get more tens?
 One way to figure this out, 
 is to find the chance of not getting any more 10's
 and then subtract that from 100%
 
 There are two more tens in the 47 cards that are not in your hand.
 That makes 45 of those cards something other than 10's.
 The chance of getting one of those other cards next is ...
 

 
 If that next card you get is not a ten, 
 there will still be 2 tens out there and 46 cards total.
 That means 44 of them will be cards that are not tens.
 The chance of getting one of the "not a ten" cards next is ...
 

 
 so the chance that neither of those two cards
 is a ten is ...
 

 
 Now there are 45 cards left, and 2 of them are tens.
 That means 43 of them are not tens.
 The chance of getting one of the "not tens" now is ...
 

 
 and the chance that none of the three cards will be tens is ...
 

 
 There is a 90% chance that you will not get any more tens,
 so there is a ...
 

 100% - 90% = 10%

 
 10% chance of getting 1 or 2 more tens.
 
 How much of that 10% is the chance of getting exactly 1 more ten?
 How much of that 10% is the chance of getting exactly 2 more tens?
 
 To answer this, 
 figure out the chance of getting 2 more tens.
 The rest of the 10%, will be the chance of getting
 only one more ten.
 
 Remember the tree diagrams?
 We can use something like the idea of one here.
 There are 45 cards of the 47 that are not tens.
 Any path that includes one of the remaining tens
 for each of two choices,
 can have any one of the 45 other cards for the third card.
 
 So, for example, there are 45 different paths 
  where the first selection is the ten of clubs, 
 and the second selection is the ten of diamonds, 
 and the third selection is one the other 45 cards. 
 
 The path types through the two other tens are ...
 
10 of Clubs 10 of Diamonds some other card 45 paths
10 of Clubs some other card 10 of Diamonds 45 paths
some other card 10 of Clubs 10 of Diamonds 45 paths
10 of Diamonds 10 of Clubs some other card 45 paths
10 of Diamonds some other card 10 of Clubs 45 paths
some other card 10 of Diamonds 10 of Clubs 45 paths
 
 There are 47 x 46 x 45 paths through the cards.
 That makes the chance of getting both of the other tens ...
 

 
 That means the chance of getting exactly one more ten is ...
 

10.0% - 0.28% = 9.72%

 
 The last strategy was to try to get the Royal Flush.
 To do this, we give up the 6 of Hearts and the10 of Spades
 and try to get the King and Ace of Hearts.
 
 What is the chance that the two cards that you get, 
 will be these two cards? 
 
 There are two cards out there you want.
 There are 47 cards out there total.
 The chance that the first card you get
 is one of the two that you want, is ...
 

 
 If we do get it, we will still need the other one.
 There are now 46 cards out there.
 The chance of getting the one we need from the 46 is ...
 

 
 and the chance of getting both of the cards we need is ...
 

 
 Let's Review:
 
 You start with ...
 
10 of Hearts
10 of Spades
6 of Hearts
Jack of Hearts
Queen of Hearts
 
 The chance of getting a Heart to replace the 10 of spades, 
 and give you a flush is ...
 

 
 The chance of getting 1 or 2 more tens 
 to replace the three cards that are not tens, is ...
 

 

 100% - 90% = 10% (a 1 in 10 chance)

 
 The chance that you got both of the other tens is ...
 

 

 
 So the chance that you got only one other ten is ...
 

 10.0% - 0.28% = 9.72% (still about a 1 in 10 chance)

 
 So the most likely of these 
 is the chance of getting the flush.
 But that's still only 20%
 To go for it (or the Royal Flush) you have to give up the pair of tens.
 
 The chance of getting more tens 
 is about half the chance of getting the flush.
 But to go for the tens, you don't give up much of anything ...
 

   copyright 2005 Bruce Kirkpatrick

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