像大多數男孩一樣，兒子最喜歡的當然是電子遊戲。如果大學有電子遊戲專業那一定是他的首選。另外他對各種智力型的遊戲如打牌，下棋，麻將等都非常有興趣。他三歲時就跟爺爺學下中國象棋，以後又學了國際象棋，因為沒有同伴經常一起練，也沒有請私人教練，所以水平一般，隻能算是業餘愛好。
八年級時兒子迷上了德州撲克（Texas hold’em）。當時電視上經常轉播世界撲克大賽(World Series of Poker)。他從電視上學了Poker的玩法和規則，就拉著爸爸，媽媽，和7歲的妹妹一起玩。Poker雖然玩法簡單，但要玩得好也非常不容易，你要對各種牌出現的幾率非常清楚，還要根據對手的下注（Bet）情況來判斷對手是否有好牌。兒子當時為準備Mathcounts學了簡單的概率，他把Poker中一手牌可能出現的各種牌型的幾率全部算了出來，並把計算的過程和結果寫進了初中的Math Project。如果你有興趣，可以看看下麵的Project。
兒子很快發現在家裏玩不下去了，媽媽輸了要耍賴，妹妹輸了要哭鬧，爸爸對兒子太了解了，經常能猜到他手上是否有好牌。兒子上了高中參加的第一個Club就是Poker Club。還好其他同學沒有他那麽著迷，玩了一年後Poker Club就解散了。
兒子在學AP Economics時，發現很多概念如競爭，壟斷，寡頭壟斷等和Poker中的許多概念很類似。老師在講解寡頭壟斷時引進了博弈論（Game Theory）是他最感興趣的。各種智力型的遊戲如打牌，下棋，麻將等都運用博弈理論。另外兒子擅長的概率統計在經濟學中也有廣泛的應用。雖然兒子在申請大學前還不清楚自己到底要學什麽，最後根據老師的推薦選了經濟學，他的選擇還是非常符合他的興趣和特長的。
Poker Hands
Poker is a very popular game today especially on TV. I have found the probability of drawing any hand in straight poker. To figure out the probability, I used the number of final hands that are favorable divided by the total number of final hands possible. For all of these calculations, the possible number of outcomes will always be 52C5 (2,598,960). An easier way of seeing this is that in straight poker, you must draw exactly five cards. In a standard deck there are 52 cards. So, the first card you can choose from 52, the second you can choose from 51, and so on until you choose your last card out of 48. You would find the product of 52, 51, 50, 49, and 48 to find all of the possibilities. Now you might be saying that this is 311,875,200, not 2,598,960. However, this is all of the possibilities and in poker, it does not matter in what order you draw the five cards. To find the number of final hands I divided 311,875,200 by 120 because there are 120 ways of rearranging each hand. This results in 2,598,960, the total number of final hands in straight poker and a number you will see billions of times in this paper.
I’ll start with the highest hand and go down to the lowest. Since I already stated that the total number of final hands in straight poker is 2,598,960, I will just find the number of favorable outcomes. The highest hand in straight poker is the Royal Flush. It contains Ace, King, Queen, Jack, and Ten all in the same suit. I put all of my equations on a separate piece of paper if you prefer to see them that way. First of all, there is only one number combination to getting a Royal Flush (the numbers, not including the suits). Second, is looking at the suits. There are four suits in the entire deck (hearts, diamonds, spades, and clubs). So, there is one number combination and four suits to choose from so I did 1*4=4 to find the number of favorable outcomes. When I put that number over 2,598,960 and simplified, I found that the probability of drawing such a hand is 1 out of 649,740 (just over .0001%). So if you decided to waste you college years by playing straight poker one hand every three minutes non-stop every day, you still probably will not get a Royal Flush by the time you graduate.
The next highest hand is the Straight Flush. This hand actually includes the Royal Flush so I’ll include that in as well. The Straight Flush is a straight (consecutive numbers) and a flush (all the same suit) at the same time. One example is 2,3,4,5, and 6 that are all spades. First off, I calculated the number combinations for a Straight Flush. There are 10 straight combinations for poker (starting with 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace). Next, just like before, there are four suits to choose your Straight Flush to be. So, I multiplied 10*4=40 to find the number of favorable outcomes. When I put that number over 2,598,960 and simplified, I found that the probability of drawing such a hand is 1 out of 64,974 (just over .001%).
After the Straight Flush is Four of a Kind. It is just what it sounds like, Four of a Kind (2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, King, or Ace) and another card known as the kicker. So you could have four 2s and a King or four Kings and a 2, and they would be considered different. First, I calculated the Four of a Kind part. There are 13 number cards to choose from to be your Four of a Kind. Since this hand uses Four of a Kind, we do not need to worry about which suits we are picking. Next, after taking the Four of a Kind, there are 48 cards left to choose from to be your kicker. So, to find the total number of favorable outcomes I multiplied 13*48=624. I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 1 out of 4,165 (about .024%).
The next highest hand is known as the Full House. A Full House actually consists of a Three of a Kind and a Pair. Three 2s and two Kings would be different from three Kings and two 2s. I separated this into finding the possibilities of getting a Three of a Kind multiplied by the possibilities of getting a Pair. I assumed that the Three of a Kind would be the first group of cards in my hand. You can choose from 13 numbers for your Three of a Kind. Now, since this is a Three of a Kind, we have to worry about the suits of all three of the cards. There are four ways of choosing three suits out of the four possible. So there are 13*4=52 ways of picking the Three of a Kind. Next is finding the Pair. Now that I have already chosen one number for the Three of a Kind, there are 12 numbers left to choose from for the Pair (No we cannot have a Three of a Kind and a Pair that is the same number). We also have to worry about the suits for the Pair as well. There are total of six ways of choosing two suits out of the four possible. So, the total number ways of picking the remaining Pair is 12*6=72. Now to find the number of possibilities for a Full House I multiplied the possibilities of the Three of a Kind with the possibilities of the Pair. I did 52*72=3744 to find the total number of favorable outcomes. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 6 out of 4,165 (about .144%).
After the Full House is the Flush. As I explained earlier, a flush consists of five cards that have the same suit. One example of a Flush is 2, 5, 8, Jack, and Ace that are all clubs. First off I looked at the numbers in the hand. The total number of possibilities for the numbers in a flush is 1,287. You can think of it this way, there are 13 ways of picking the first number, 12 for the second, and so on until you reach 9 for the last card. However, as in the first paragraph, there are 120 ways of picking up the five cards so you divide the product of 13*12*11*10*9 by 120 to get 1,287. Next, there are four suits to choose from for Flush. I multiplied 1,287*4=5,148 to find the total number of ways of getting a Flush. But, since I am calculating a Flush, I cannot include the number of ways of drawing a Straight Flush. So, I subtracted 40 (look at the paragraph about Straight Flush) from the total of 5,148 (and got 5,108) to find the total number of favorable outcomes for just a Flush. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 1,277 out of 649,740 (about .197%). This hand can be compared to winning a raffle in the whole eighth grade. You would have to be the lucky one person to win out of five hundred.
The next highest hand is a Straight. As I explained earlier, a Straight consists of five consecutive numbers. One example of a Straight is 2, 3, 4, 5, and 6 in which not all of the cards are the same suit. First off, I calculated the number combinations for a Straight. There are 10 straight combinations for poker (starting with 5, 6, 7, 8, 9, 10, Jack, Queen, King, and Ace). Next, I calculated all of the suits that the numbers could consist of. I said that each card in whatever Straight drawn has four suits to choose from. So, I said that there are 4*4*4*4*4=1,024 ways of choosing the suits for the cards to be. Then, I multiplied 10*1,024 to find the number of favorable outcomes. However, just like before, I needed to subtract the number of ways of drawing a Straight Flush because I am calculating the probability of a Straight. So, I subtracted 40 (look at the paragraph about Straight Flush) from the total of 10,240 (and got 10,200) to find the total number of favorable outcomes for just a Straight. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 5 out of 1,274 (about .392%).
After a Straight is Three of a Kind. A Three of a Kind is just like a Four of a Kind except now there is only Three of a Kind and two kickers, which must be different numbers. One example of a Three of a Kind is three Kings, 4, and Ace. First I calculated the Three of a Kind part. You can choose from 13 numbers for your Three of a Kind. Now, since this is a Three of a Kind, we have to worry about the suits of all three of the cards. There are four ways of choosing three suits out of the four possible. So there are 13*4=52 ways of picking the Three of a Kind. Next, I calculated the possibilities for the two kickers. There are 48 ways of choosing the fourth card and 44 ways of choosing the last card. This is because the fourth card cannot correspond with the Three of a Kind, and the last card cannot correspond with the Three of a Kind or the fourth card. So there are 48*44=2,112 ways of picking the last two cards. However, I am calculating the possibilities for final hands and the last two kickers can be reversed. One example is, assume that you picked up three Aces, 2, and 3 on your first draw. Then, on the next turn you pick up three Aces, 3, and 2. That is the same thing, but by multiplying 48*44=2,112, the order in drawing does matter (the 2 could be from the 48 ways and the 3 could be from the 44 ways or vice versa) (I hope I made sense). Since the last two numbers can be interchanged and still be the same hand so I divided 2,112 by 2 and got 1,056 final ways of choosing the kickers. So, I multiplied 52*1,056=54,912 to get the total number of favorable outcomes. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 88 out of 4,165 (about 2.11%).
The next highest hand is Two Pairs. This consists of; well you probably guessed it, Two Pairs. One example of Two Pairs is two Aces, two Kings, and 5. First I calculated the Two Pairs. There are 13 numbers that you can choose from for your first Pair. Just like before, there are six ways of choosing two suits from the four possible. So, there are 13*6=78 total ways of getting the first Pair. Next, I calculated the possible ways of drawing the second Pair. Since the first Pair already uses up one of the 13 numbers, there are only 12 left to choose from. Again there are six ways of choosing two suits from the four possible. So there are 12*6=72 total ways of getting the second Pair. So there are 78*72=5,616 total ways of drawing the Two Pairs. However, the Two Pairs can be reversed, just like the two kickers in the Three of a Kind can be reversed (look in the paragraph above to see why). So I divided 5,616 by 2 and got 2,808 final ways of drawing the Two Pairs. Then, I found that there are 44 ways of drawing the last card because it cannot correspond to either of the Two Pairs. Next, I multiplied 2,808*44=123,552 to get the total number of favorable outcomes. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 198 out of 4,165 (about 4.75%).
After Two Pairs is one Pair. This consists of; do I even have to tell you, one Pair. One example is two Aces, 2, 3, and 6. First I calculated the possibilities for the Pair. There are 13 numbers that you can choose from for your Pair. Just like before, there are six ways of choosing two suits from the four possible. So, there are 13*6=78 total ways of getting the Pair. Next, I calculated the possibilities for the three extra cards. I did 48*44*40=84,480 to get how many ways there are to draw the three remaining cards. This is because each card cannot correspond to the Pair or any of the other remaining cards. However, the three extra cards can be rearranged in six ways so I divided 84,480 by 6 and got 14,080 distinct ways of getting the last three cards (to see the full explanation of why, look two paragraphs above; it is the same idea except with three cards instead of two). Next, I multiplied 14,080*78=1,098,240 to get the total number of favorable outcomes. Again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 352 out of 833 (about 42.26%).
The lowest hand possible is having none of the hands above (I call it a Junk Hand). All you have to do is take 2,598960 and subtract all favorable outcomes above. When you do this you will get 1,302,540 favorable outcomes for the Junk Hand. Yet again, I put this number over 2,598,960 and simplified it, finding that the probability of drawing such a hand is 1,277 out of 2,548 (about 50.12%). So in a game of Straight Poker between two people, there will almost always be one good hand and one Junk Hand.
The final analysis between the Poker Hands is that the higher the probability of drawing the hand, the lower it is considered in terms of value. The lower the probability of drawing a hand, the higher it is considered in terms of value. Also, the probability of getting a higher hand decreases exponentially each time you move to the next highest hand.
