Numb3rs

Season 5

Episode 12: Jacked

A recurring reference in this episode is the game of poker. Charlie compares the hostage situation to knowing an opponents betting strategy.

What is Poker?

In the most simple variety of poker, a number of players are dealt five cards from a standard 52 card deck which consists of four suits, ♠, ♥, ♦, ♣ of cards labeled 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. These five cards make a "hand." Each player then makes a bet as to whether he or she has the best hand, where "best" will be defined as follows. The value of each card is ranked in ascending order: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. There are then several different possibilities for hands. A hand is a "flush" if all five cards are of the same suit. A hand is a "straight" if the cards are in consecutive order (meaning 3, 4, 5, 6, 7, for example). A straight cannot "wrap around" (meaning J, Q, K, A, 2 is not a straight). However, A, 2, 3, 4, 5 violates this rule and is considered a straight. A "straight flush" is a hand which is a straight and a flush. A hand can have a pair of cards (two Ks, for example), two pairs of cards (two Ks and two Qs, for example), "three of a kind" (three As, for example), "four of a kind" which consists of four of a particular number (four As, for example), or a "full house" which consists of a three-of-a-kind as well as a pair (three As and two Ks, for example). If a hand is none of these things, the highest card is the most important feature (for example, 2♠ 4♥ 5♥ 10♣ K♣ is called king high). The hands are then ranked as follows, in ascending order:

One may wonder why the hands are ordered in this fashion. This is because there are fewer hands of a given type as one moves up the list from #9 to #1. See below.

How many Straight Flushes are possible?

To make a straight flush, one has to pick a suit. There are four choices for that. Then one has to pick the lowest card in the straight for which there are ten choices: A, 2, 3, 4, 5, 6, 7, 8, 9, 10. Hence there are

straight flushes possible.

How many Four of a Kinds are possible?

There are 13 choices for the card in the four of a kind. The remaining card can be freely chosen, so there are 48 of them. Hence there are

four of a kinds possible.

How many Full Houses are possible?

One has to pick the the card involved in the three of a kind and the card involved in the pair. There are 13 choices for the first and 12 choices for the second. For the three of a kind, one has four suits to choose from and we need to pick three. There are four ways to do that. For the pair, there are four suits and we need to pick two of them. There are six ways to do that. So there are

possible full houses.

How many Flushes are possible?

To form a flush, we have to pick one of the four suits. Then one has to pick 5 out of the 13 cards of that suit. This is given by 13!/(5!8!)=1287. So there are

possible flushes. However,
this includes the 40 straight flushes, which means there are really
only 5108 flushes.

How many Straights are possible?

To form a straight, one needs to pick the first card in the straight (this can be one of A, 2, 3, 4, 5, 6, 7, 8, 9, 10 of any suit). So, there are 40 possible first cards. The value of each consecutive card is then fixed but each can be of any of the four suits. Hence there are

different straights possible. This includes 40 straight flushes, so there are really only 10200 straights.

How many Three of a kinds are possible?

To form a three of a kind, one has to pick one of the 13 possible card values. One then has to pick three of the four suits for that value. There are four ways to do this. To avoid a four of a kind and a full house, I have to pick 2 different values for the remaining two cards, given by 12!/(10!2!)=66. For each of these cards, there are four suits. Hence there are

three of a kind possible.

How many two pair hands are possible?

To construct a two pair hand, we have to pick two values for the pairs. There are 13!/11!2!=78 ways to do this. For each pair, we need to pick two of the four suits. There are 6 ways to do this for each pair. For the final card, we can pick any of the remaining 44 cards. Thus there are

possible two pair hands.

How many one pair hands are possible?

To form a hand with just one pair, we must pick a value for the pair. There are 13 of these. We must also pick two suits from the four possible, which can happen in 6 different ways. For the remaining three cards, each value must be distinct and different from the value of the pair. There are 12!/3!9!=220 ways to do this. For each of these cards, we can pick any of the four suits. Hence there are

different possible hands which have a single pair and aren't full houses or four of a kind.

How many hands are just a high card hand?

The easiest way to determine this is to take number of all the possible hands and subtract from that the number of hands above. To pick any hand, we must pick 5 cards from 52, of which there are 52!/47!5!=2598960 possibilities. Hence there are

Nearly no one actually plays the version of poker described above. There are a number of different variants of poker, the most popular of which fit into Stud and Hold'em varieties.

Hold'em!

Hold'em Poker comes in essentially two varieties: Texas and Omaha. Both can involve up to 8 players. In both varieties, each player is dealt some cards face down (the player may look at their cards). During successive rounds of betting, community cards are dealt face up into the middle of the table which each player uses, along with their down cards, to make a five card hand. In Texas Hold'em, each player is dealt two cards face down. The rounds of betting which follow are:

In Omaha, each player receives four cards face down. The betting and the way the cards come out is exactly the same as in Texas Hold'em, except for how players may form hands. They must use exactly two of the cards in their hand and exactly three of the cards on the board. In the standard form of Omaha, players make the best possible five card hand, following the rules above. In a second (and very common) form of Omaha called Omaha Hi-Lo, each player forms the best five card poker hand and the worst five card poker hand they can. The amount of money bet is split between the player with the best hand and the player with the worst hand (this can be the same player). A subtle rule is that the low hand must be 8-high or lower. The low-hands often tie as a result of this rule.

Stud Poker.

In Stud poker variants, each player is dealt a sequence of face-up and face-down cards after rounds of betting. The number of cards varies, although Seven-Card Stud is the most common. Seven-Card stud typically proceeds as follows:

Suppose that one were to program a computer to play the simplest variant of poker with a very conservative set of rules. Typically, there is a minimum bet required which we'll call B (some small but not too small number which depends on how much money everyone starts playing with). Suppose our computer would calculate the exact probability that its particular hand wins against a random hand. Call this probability p. Suppose that the machine always makes a bet equal to B/(1-p) (or as much as it can if this number is too large). So, if p is zero, it makes the minimum bet. As p tends to one, it bets more and more. One may wonder how it would be possible to beat such a machine.

If one were to play against the machine for some time, one would notice that it never bets much when it has a weak hand. But when it has a strong hand, it seems willing to bet a lot. So what strategy should you adopt? If you play conservatively, meaning you adopt a similar strategy you will probably lose over the long run --- you will make mistakes computing the odds while the computer will not. However, you can bluff while the computer is incapable of doing so. So when the computer bets a relatively high amount but not that much, you can come over the top with a huge bet which will force the computer to fold because it rigidly adheres to the odds. So knowing your opponents strategy ahead of time gives you a huge advantage.

This is the key to playing poker: information gained from how your opponent bets. In Texas Hold'em before the flop, suppose that a player bets a significant amount. Chances are, he or she has a relatively high pair or something like AQ. He or she doesn't want players with low suited cards or middle consecutive cards to see the flop. If the flop then comes up with lower cards in sequence or of the same suit, a player, even if he or she didn't hit anything on the flop, can make a large bet which a pre-flop high pair or AQ has to respect because they didn't hit an ace or a king, and there are plenty of better hands possible. The key to making such a bet is to bet enough to get the other player to fold but not enough to make it seem like that's exactly what you want. As Han Solo in Return of the Jedi says to his copilot, "Keep your distance... but don't look like you're keeping your distance. I don't know, fly casual." Of course this is much easier said than done.

If several players call a small bet before the flop, hands which are generally not great become better. The more players get into the pre-flop bet, the more even the game becomes. Low consecutive cards and cards of the same suit become more valuable (because they may be unlikely winners in general, but if they hit, they win a lot with so many players betting) while high cards lose their luster because several players are likely to pair up (and so their individual value plummets).

As play continues, the first person to bet rotates from player to player. Being the last person to bet allows you to collect all the possible information before you make your bet. If a couple of players are staying in the hand and no one is making a strong move, the last person to bet can make a move to push them into folding or to make it seem that his or her hand is stronger than it really is. One can also "limp" into the hand, meaning one has a very good hand but wants to make it seem weaker than it is so opponents will feel their hand is strong enough to bet with.

There is also a huge difference between how one strategizes when comparing Texas Hold'em and Seven-card Stud. In Hold'em, you only get to see two cards before the first round of betting. In Stud, you get to see your two down cards and your up card, not to mention the up card of every opponent. The amount of information in the game is drastically increased during each round of betting. An experienced player can take in all the information at the table. If one of his opponents is showing a 9 and a 10, then he or she knows that there are fewer straights available to him/her which utilize the 9 and 10. This is one of the reasons Stud is less popular than Hold'em: it is much more complicated. A new player can get relatively proficient in how to play Hold'em in a short period of time. New information in Hold'em can get absorbed slowly, even if the player is not necessarily spectacular at utilizing it. In Stud, the information comes extremely quickly and is thus difficult to synthesize, even incorrectly. It takes quite a bit longer to get used to that. Even professional level skill at Hold'em does not carry over exceptionally well into Stud.

Season 5

Episode 12: Jacked

A recurring reference in this episode is the game of poker. Charlie compares the hostage situation to knowing an opponents betting strategy.

What is Poker?

In the most simple variety of poker, a number of players are dealt five cards from a standard 52 card deck which consists of four suits, ♠, ♥, ♦, ♣ of cards labeled 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. These five cards make a "hand." Each player then makes a bet as to whether he or she has the best hand, where "best" will be defined as follows. The value of each card is ranked in ascending order: 2, 3, 4, 5, 6, 7, 8, 9, 10, J, Q, K, A. There are then several different possibilities for hands. A hand is a "flush" if all five cards are of the same suit. A hand is a "straight" if the cards are in consecutive order (meaning 3, 4, 5, 6, 7, for example). A straight cannot "wrap around" (meaning J, Q, K, A, 2 is not a straight). However, A, 2, 3, 4, 5 violates this rule and is considered a straight. A "straight flush" is a hand which is a straight and a flush. A hand can have a pair of cards (two Ks, for example), two pairs of cards (two Ks and two Qs, for example), "three of a kind" (three As, for example), "four of a kind" which consists of four of a particular number (four As, for example), or a "full house" which consists of a three-of-a-kind as well as a pair (three As and two Ks, for example). If a hand is none of these things, the highest card is the most important feature (for example, 2♠ 4♥ 5♥ 10♣ K♣ is called king high). The hands are then ranked as follows, in ascending order:

- High card (if two players have the same high card, the next
highest card is compared, then the next highest, and so on)

- One pair (if two or more players have the same pair, they compare
the highest of their remaining three cards, then the next highest, then
the next highest)

- Two pair (if two more more players have two pair, the player with
the highest pair wins; if two players have the same high pair, the
player with the next highest pair wins; if both pairs are the same,
compare the fifth card)

- Three of a kind (higher three of a kinds are better)

- Straight (the higher the value of the highest card in a straight,
the better the straight)

- Flush (in the case of two or more flushes, the flush with the
highest card in it wins; if necessary, compare next highest cards, etc.)

- Full house (if multiple players have full houses, the better
three of a kind wins)

- Four of a kind (if more than one player has four of a kind, the
higher four of a kind wins)

- Straight Flush (the better straight wins if more than one player
has a straight flush)

One may wonder why the hands are ordered in this fashion. This is because there are fewer hands of a given type as one moves up the list from #9 to #1. See below.

How many Straight Flushes are possible?

To make a straight flush, one has to pick a suit. There are four choices for that. Then one has to pick the lowest card in the straight for which there are ten choices: A, 2, 3, 4, 5, 6, 7, 8, 9, 10. Hence there are

How many Four of a Kinds are possible?

There are 13 choices for the card in the four of a kind. The remaining card can be freely chosen, so there are 48 of them. Hence there are

four of a kinds possible.

How many Full Houses are possible?

One has to pick the the card involved in the three of a kind and the card involved in the pair. There are 13 choices for the first and 12 choices for the second. For the three of a kind, one has four suits to choose from and we need to pick three. There are four ways to do that. For the pair, there are four suits and we need to pick two of them. There are six ways to do that. So there are

How many Flushes are possible?

To form a flush, we have to pick one of the four suits. Then one has to pick 5 out of the 13 cards of that suit. This is given by 13!/(5!8!)=1287. So there are

How many Straights are possible?

To form a straight, one needs to pick the first card in the straight (this can be one of A, 2, 3, 4, 5, 6, 7, 8, 9, 10 of any suit). So, there are 40 possible first cards. The value of each consecutive card is then fixed but each can be of any of the four suits. Hence there are

different straights possible. This includes 40 straight flushes, so there are really only 10200 straights.

How many Three of a kinds are possible?

To form a three of a kind, one has to pick one of the 13 possible card values. One then has to pick three of the four suits for that value. There are four ways to do this. To avoid a four of a kind and a full house, I have to pick 2 different values for the remaining two cards, given by 12!/(10!2!)=66. For each of these cards, there are four suits. Hence there are

How many two pair hands are possible?

To construct a two pair hand, we have to pick two values for the pairs. There are 13!/11!2!=78 ways to do this. For each pair, we need to pick two of the four suits. There are 6 ways to do this for each pair. For the final card, we can pick any of the remaining 44 cards. Thus there are

How many one pair hands are possible?

To form a hand with just one pair, we must pick a value for the pair. There are 13 of these. We must also pick two suits from the four possible, which can happen in 6 different ways. For the remaining three cards, each value must be distinct and different from the value of the pair. There are 12!/3!9!=220 ways to do this. For each of these cards, we can pick any of the four suits. Hence there are

different possible hands which have a single pair and aren't full houses or four of a kind.

How many hands are just a high card hand?

The easiest way to determine this is to take number of all the possible hands and subtract from that the number of hands above. To pick any hand, we must pick 5 cards from 52, of which there are 52!/47!5!=2598960 possibilities. Hence there are

2598960 −
1098240 − 123552 − 54912 − 10200 − 5108 − 3744 − 624 − 40 = 1302540

different hands which are just a high card.

How good is a particular hand?

The useful thing to know is how likely I am to win if I have a particular hand. For example, suppose I have a pair of kings. How likely am I to win? This is actually a relatively complicated problem to calculate exactly. For example, it makes certain straights and flushes less likely. But these are a small percentage of the total number of hands. Two kings beats or ties all but one of the one pair hands and all of the high card hands. This amounts to (12/13)(1098240) + 1302540 = 2316300, or roughly 89% of hands. So I'm in good shape. A pair of twos beats or ties only (1/13)(1098240) + 1302540 = 1387020, or roughly 53% of hands. That is not a hand to be extremely confident about.

Variants of Poker.

different hands which are just a high card.

How good is a particular hand?

The useful thing to know is how likely I am to win if I have a particular hand. For example, suppose I have a pair of kings. How likely am I to win? This is actually a relatively complicated problem to calculate exactly. For example, it makes certain straights and flushes less likely. But these are a small percentage of the total number of hands. Two kings beats or ties all but one of the one pair hands and all of the high card hands. This amounts to (12/13)(1098240) + 1302540 = 2316300, or roughly 89% of hands. So I'm in good shape. A pair of twos beats or ties only (1/13)(1098240) + 1302540 = 1387020, or roughly 53% of hands. That is not a hand to be extremely confident about.

Variants of Poker.

Nearly no one actually plays the version of poker described above. There are a number of different variants of poker, the most popular of which fit into Stud and Hold'em varieties.

Hold'em!

Hold'em Poker comes in essentially two varieties: Texas and Omaha. Both can involve up to 8 players. In both varieties, each player is dealt some cards face down (the player may look at their cards). During successive rounds of betting, community cards are dealt face up into the middle of the table which each player uses, along with their down cards, to make a five card hand. In Texas Hold'em, each player is dealt two cards face down. The rounds of betting which follow are:

- After players receive their face-down cards, there is a round of betting.
- Three community cards are dealt (called "the flop"), and there is another round of betting.
- A fourth community cards is dealt (called "the turn"), and there is another round of betting.
- A fifth community cards is dealt (called "the river") with a final round of betting.
- Remaining players show their face-down cards and make the best possible five-card hand using zero, one, or two of their face-down cards and three, four, or five community cards. This can sometimes result in a tie, in which case the tying players split the money bet.

In Omaha, each player receives four cards face down. The betting and the way the cards come out is exactly the same as in Texas Hold'em, except for how players may form hands. They must use exactly two of the cards in their hand and exactly three of the cards on the board. In the standard form of Omaha, players make the best possible five card hand, following the rules above. In a second (and very common) form of Omaha called Omaha Hi-Lo, each player forms the best five card poker hand and the worst five card poker hand they can. The amount of money bet is split between the player with the best hand and the player with the worst hand (this can be the same player). A subtle rule is that the low hand must be 8-high or lower. The low-hands often tie as a result of this rule.

Stud Poker.

In Stud poker variants, each player is dealt a sequence of face-up and face-down cards after rounds of betting. The number of cards varies, although Seven-Card Stud is the most common. Seven-Card stud typically proceeds as follows:

- Three cards are dealt to each player, two of which are face down
and one face up. There is a round of betting.

- One card is dealt to each remaining player face up. There is a round of betting.
- One card is dealt to each remaining player face up. There is a round of betting.
- One card is dealt to each remaining player face up. There is a round of betting.
- One card is dealt to each remaining player face down. There is a round of betting.
- The remaining players show all their cards, making their best
five-card poker hand possible using the seven cards they have been
dealt.

Suppose that one were to program a computer to play the simplest variant of poker with a very conservative set of rules. Typically, there is a minimum bet required which we'll call B (some small but not too small number which depends on how much money everyone starts playing with). Suppose our computer would calculate the exact probability that its particular hand wins against a random hand. Call this probability p. Suppose that the machine always makes a bet equal to B/(1-p) (or as much as it can if this number is too large). So, if p is zero, it makes the minimum bet. As p tends to one, it bets more and more. One may wonder how it would be possible to beat such a machine.

If one were to play against the machine for some time, one would notice that it never bets much when it has a weak hand. But when it has a strong hand, it seems willing to bet a lot. So what strategy should you adopt? If you play conservatively, meaning you adopt a similar strategy you will probably lose over the long run --- you will make mistakes computing the odds while the computer will not. However, you can bluff while the computer is incapable of doing so. So when the computer bets a relatively high amount but not that much, you can come over the top with a huge bet which will force the computer to fold because it rigidly adheres to the odds. So knowing your opponents strategy ahead of time gives you a huge advantage.

This is the key to playing poker: information gained from how your opponent bets. In Texas Hold'em before the flop, suppose that a player bets a significant amount. Chances are, he or she has a relatively high pair or something like AQ. He or she doesn't want players with low suited cards or middle consecutive cards to see the flop. If the flop then comes up with lower cards in sequence or of the same suit, a player, even if he or she didn't hit anything on the flop, can make a large bet which a pre-flop high pair or AQ has to respect because they didn't hit an ace or a king, and there are plenty of better hands possible. The key to making such a bet is to bet enough to get the other player to fold but not enough to make it seem like that's exactly what you want. As Han Solo in Return of the Jedi says to his copilot, "Keep your distance... but don't look like you're keeping your distance. I don't know, fly casual." Of course this is much easier said than done.

If several players call a small bet before the flop, hands which are generally not great become better. The more players get into the pre-flop bet, the more even the game becomes. Low consecutive cards and cards of the same suit become more valuable (because they may be unlikely winners in general, but if they hit, they win a lot with so many players betting) while high cards lose their luster because several players are likely to pair up (and so their individual value plummets).

As play continues, the first person to bet rotates from player to player. Being the last person to bet allows you to collect all the possible information before you make your bet. If a couple of players are staying in the hand and no one is making a strong move, the last person to bet can make a move to push them into folding or to make it seem that his or her hand is stronger than it really is. One can also "limp" into the hand, meaning one has a very good hand but wants to make it seem weaker than it is so opponents will feel their hand is strong enough to bet with.

There is also a huge difference between how one strategizes when comparing Texas Hold'em and Seven-card Stud. In Hold'em, you only get to see two cards before the first round of betting. In Stud, you get to see your two down cards and your up card, not to mention the up card of every opponent. The amount of information in the game is drastically increased during each round of betting. An experienced player can take in all the information at the table. If one of his opponents is showing a 9 and a 10, then he or she knows that there are fewer straights available to him/her which utilize the 9 and 10. This is one of the reasons Stud is less popular than Hold'em: it is much more complicated. A new player can get relatively proficient in how to play Hold'em in a short period of time. New information in Hold'em can get absorbed slowly, even if the player is not necessarily spectacular at utilizing it. In Stud, the information comes extremely quickly and is thus difficult to synthesize, even incorrectly. It takes quite a bit longer to get used to that. Even professional level skill at Hold'em does not carry over exceptionally well into Stud.