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Poker MathematicsFor studying mathematical models which illustrate key principles of potlimit hold’em poker play, London lowball is an excellent game. The reasons are the last card dealt to each poker player is not visible to the other poker players (as opposed to games with common cards like hold’em and Omaha), and with one card to come, it can be clear that a hand is presently behind. However, it will surely win if it improves. We will discuss the two related ideas in action. First, proper bluffing frequency; second, the leverage on the end that a drawing hand possesses. If it busts out, how often should a drawing hand bluff? Obviously, this is a decision with strong psychological elements. However, we should understand that if those elements are removed, mathematics can dictate the proper frequency and the rival poker players in the pot play in a “correct” manner which means at random and with the right frequency of folding and calling. According to poker game theory, we should always make the decisions on the last betting round to determine the proper frequency of poker bluffing as always calling or always calling yield the same result. We will assume that the poker player will always bet precisely what is in the pot. It will not matter in the long run how the rival poker player reacts to our bet as he will get the same result if we bluff exactly half as often as the frequency of betting with a winning hand. To illustrate this principle, let us use a concrete example. Assume that there is a grand in the pot going into seventh street. The best hand is with poker player A and poker player B will have the best hand in all cases where he improves on the last card. Obviously, poker player A should check at the river. If poker player B has ten winning cards he can hit. Poker player B is bluffing the exact amount and the strategy adopted by poker player A is not related if poker player B bluffs on five of the cards where he misses. The result will always be the same. Poker player A would not lose any additional money if he always folds. Poker player A will lose $10000 on the hands where poker player B has hit if poker player A always calls. However, poker player A will gain back $10000 on the five cards where poker player B has run a bluff ( $5000 originally in the pot, plus $5000 in poker player B’s bluffs). We can have look at the scenario in a different way as poker player A who gets 21 on the money will break even by always calling when poker player B bets on the end using the proper poker game strategy of bluffing half as often as having the goods. Despite bluffing, poker player B is doing well on these fifteen hands, profiting by winning an equivalent amount to the main pot. A good poker player is going to do better in our mathematical model. By noticing whether his rival poker player is normally a caller or a folder, he will vary from the dictates of the poker game theory and whether he has a tell when bluffing, and so forth. To tell us when the poker player with the best hand should check going into the last card to deprive the rival poker player of the extra leverage gained from the bigger pot size, here is another model. For e.g. two poker players are contesting a pot with a grand in it. As in the previous scenario, poker player A is leading with one card to come, and poker player B is trying to draw out. Assume there is enough money left for a bet and raise, or a bet now and a bet on the end. This amount would be four thousand apiece for a grand in the pot. How much is required by poker player A for it to be correct to bet at the point where there is one card to come? Always keep in mind that the bet now triples the amount that poker player B can bet on the end. When you are increasing the leverage for the drawing hand on the end, is betting a good idea? Let us see where the breakeven point between checking and betting is located. That point is when poker player B has exactly a third of the available cards with which to improve. By understanding what happens in our thousand dollar pot when there are 30 unknown cards, we illustrated this. Poker player B draws our with 10 of them, and poker player A has his hand stand up with the other 20. ( There would be more unknown cards than this in most online poker games. However, these numbers are easier to work with as the ratios are the same.) Poker player B will bet on his 10 winning cards and 5 of his losing cards. Poker player A is going to break even on his action in the case where he checks with one card to come and always calls on the end. On five hands, poker player A makes a profit of $2000 * 5= $10,000 where poker player B bluffs. In addition to this, he also makes a profit of $1,000 * 15= $15,000 on the fifteen hands where poker player B busts out and does not attempt to bluff. Therefore, poker player A has a total profit of $25,000 on all the pots that he wins. Poker player A loses just 10 * $1,000= $10,000 on the ten pots which he lost. 
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