SOME INTERESTING BUT NOT TOO USEFUL MATHEMATICAL FACTS:

In hold'em, with no betting after the flop, all pairs are a favorite over Ace-King oflsuit, by at least 529 to 489. All pairs except Deuces with neither card in the Ace-King's suit are favored over Ace-King suited. If several players lold first, Ace-King suited is a favorite over most pairs. (The exceptions are Aces, Kings, and Jacks, and also lens where one of the lens is the same suit as the Ace-King.) Even Ace-King oflsuit is now a favorite against small pairs. The reason for this is that players are more likely to play hands having an Ace or King than those containing smaller cards. Therelore, as players lold, the probability of an Ace or King coming on the board increases.

The chances of having pocket Aces in hold'em are I in 221. When everyone folds to you and you have the big blind, the probability of having a pair of Aces is approximately 1 in 134, since players are more likely to enter the pot if they have an Ace.

Similarly, but more obviously, when everyone folds to the big blind in lowball, it is likely that the big blind will pick up a hand mainly consisting of low cards.

When playing heads-up triple-draw lowball, it shouldn't be surprising that pat hands gel beat on the third draw much more frequently than in single draw, since some useless cards have been discarded on the first two draws, and the opponent is draw ing to better hands.

Two players play a heads-up session for eight hours. Let's suppose they play 40 hands per hour for a total of 320 hands. If every hand has a 509c chance to be won by one player or the other, what is the average longest winning streak either of them w ill have?

If n is the number of hands, the average longest streak is approxi­mately log, ii_y even for small values of n. VVe should expect, on average, that one player will win eight hands in a row over the course of 320 hands. People tend to expect things to even out sooner than they do, so they are surprised by the streaks that come about. This leads them to believe that other things must be influencing the results: "He's just hot," "That's his lucky dealer," etc.

Other gambling games:

If you are in a gambling situation where you have the worst of it, no betting strategy will make you a lavorite. For example, there is no betting system to beat the casinos at craps.

You can play with a betting strategy that will allow you to win a high percentage ol the time, but in the long run you will still lose. You will have many small wins, but your big losses will be more than the sum of the small wins. The typical strategy used is akin to what is called a Mar­tingale Strategy

Definition of Martingale Strategy: Make a small bet. If you w in, quit. Ii you lose, bet the amount you are behind plus enough for a small profit. Repeat the process until you win.

Many players manage their poker sessions using a Martingale ap­proach. They hit and run with small wins, but if they are losing they play until they

drop. They are psychologically fulfilled 90% of the time, but they still lose money overall.

Even though betting strategies will not counteract a mathematical dis­advantage, you may lose money to someone employing a Martingale type strategy against you. For example, let's say someone allows you to Hip a coin, and he will always call heads and will pay you 2 to 1 odds. He gets to decide the amount to bet on each Hip and how many times to Hip, each time paying 2 to 1 odds. He guarantees that he will do this with you every clay, and that he will never try to beat you out of more than $ 100 for the clay. There is no settling up until he calls it quits for the day.

His system: Bet to w in S 1 00 the first flip; if he loses, he will be down S200, so he will bet to win $300 on the next flip. If he keeps losing, he will bet an amount that will let him w in S 100. The sequence of bets w ill be $100, S300, S900, S2700, and so on. If heads comes up on the nth flip, he will have bet $100 \ 3 "' ¹¹ on that flip, and he will now quit and collect his S 100 for the day. There may be some long streaks of tails and he may get up to amounts he cannot pay, but eventually heads w ill hit and he w ill collect $100. If casinos gave unlimited credit and allowed unlim­ited bets, they would all go broke. But of course, there are credit limits and table limits.

SOME USEFUL MATHEMATICAL FACTS YOU SHOULD KNOW

Hold'em: 

If vou flop a flush or a straight draw, with no other ways of w inning, vou arc approximately a 2 to 1 underdog.

If you have a small pair and need to make trips to win, you w ill not have sufficient pot odds in limit hold'em to take a card oil on the turn.

If you have 13 outs on the Hop, you will typically he a slight underdog.

If you have 14 outs on the flop, you will typically be a slight favorite.

A big pocket pair against a small pocket pair is slightly more than a 4 to 1 favorite to win the hand.

If you have a pocket pair, you are around a 7 .5 to 1 underdog to flop a set.

If vou flop a flush or a straight draw, with no other ways of w inning, vou arc approximately a 2 to 1 underdog.

If you have a small pair and need to make trips to win, you w ill not have sufficient pot odds in limit hold'em to take a card oil on the turn.

If you have 13 outs on the Hop, you will typically he a slight underdog.

If you have 14 outs on the flop, you will typically be a slight favorite.

A big pocket pair against a small pocket pair is slightly more than a 4 to 1 favorite to win the hand.

If you have a pocket pair, you are arou nd a / .5 to 1 underdog to flop a set.