Number Theoretical Ciphers - Modern Cryptanalysis: Techniques for Advanced Code Breaking - page 24

Texas Hold 'Em has been a particularly popular poker variant for quite a while, and a rather simple one to
explain. We won't get into the betting structure, but some of the mechanics will be of interest. Also, in Texas
Hold 'Em, no suit is better than any others, so there can be ties.
Every player (usually between 2 and 10 players) is dealt two cards, which only that player sees. Next, three
community cards are placed in the middle (the flop). Then, a fourth additional card is placed in the middle as
a community card (the turn). Finally, a fifth card is added to the community cards (the river).
Each player can use the community cards, as well as the cards in his or her hand, to make the best five-card
hand possible. The player with the highest hand wins.
Texas Hold 'Em is an interesting game to analyze. It's just on the edge of computationally feasible to calcu-
late exact probabilities for many of our scenarios.
A first question we might ask is, how many different seven-card combinations can there be? Well, in our
case, since order doesn't matter, there are going to be 52 cards, and we want a set of 7 of them. This can be
calculated with
Note this doesn't even take into account the fact that suits are built equally. Many of these hands will be repeats.
How many? We'll leave this as an exercise to the reader.
As we'll see more of, later, solving cryptanalytic problems involves a lot of probability. Indeed, much of
cryptanalysis is figuring out how the probabilities in one part of a cipher affect the probabilities of another part,
thenmeasuringactualoutcomestoattempttolearninformationaboutpartsthatwecannotsee,suchastheplain-
text or the key. In many ways, this is not unlike analyzing some poker situations.
Let'sanalyze some pokerscenarios. One thing someone usually wants toknowis, what is the probability that
a player will get a pair in his or her own private hand (not in community cards) to start out with? This is usually
considered to be a good thing. To start out with, it doesn't matter who is sitting where, or how many players
there are. Since the player we are concerned with can't see anyone else's cards, then there is no information to
be had, so we can ignore the fact that other people are playing.
The first card is dealt to the player, and it can be any one of the 52 cards in play, so it won't affect the player.
But, to get a pair, the second card has to be one of the only other three cards left in the deck of (now) 51 cards
in order to pair with the first card dealt. Thus, our probability is
This means that a player can expect to receive a pair dealt at the beginning about 1 out of every 17 hands, on
average.
A situation people often see themselves in playing poker is what is called a draw — where a person needs
certain cards in order to get a stronger hand, and there are still cards to be dealt.
For example, if a person has two cards of the same suit as his own personal cards (his “down” cards), it is
easy to calculate that it is fairly improbable that the person will get a flush immediately on the flop. But what if
two of the three cards dealt are the same as the player's down cards? What is the probability then that the player
will obtain a flush?
The only information known to the player is the identities of the down cards and community cards. Knowing
this, 4 of the 13 cards in the desired suit are already in play, and 1 additional card that is not, for a total of 5 cards
in play. The player has two opportunities to get a card of the desired suit: the turn and the river cards, which will
be picking cards from 47 and 46 cards, respectively.