This guide presents a mechanism for earning money through the Poker game available through the "Casino Passport" item, first acquired at Durandal. This Poker game is heavily weighted in favor of the player, such that over time it provides a consistent and guaranteed source of cash, which is otherwise lacking in the game. This guide attempts to explain how gambling can be profitable, and the quickest way to take advantage of this game to make as much money as you want.
In addition to cold hard G, you can acquire thousands of Xenocard boosters, making it trivial to create any deck you want in a matter of under an hour.
- 1 Accessing the Poker game
- 2 How to play Poker
- 3 Poker in Xenosaga
- 4 Making lots of money
- 5 Cashing out
- 6 The mathematical evidence
Accessing the Poker game
The Poker game is first accessible when you arrive at the Durandal, after completing the Cathedral Ship. In order to access the Poker game, you must first acquire the "Casino Passport", which is available inside the Casino in the Residential Area of the Durandal.
Getting the Casino Passport
After arriving at the Durandal and being allowed to leave the Isolation Area, enter the Durandal Train. Select "Residential Area" as your destination, and enter the Train. When you exit the train, proceed westward until you come to a wall. Turn to the north, and continue down a hallway, past some vending machines. When you reach the north wall, make a right turn, and enter the first door to the north. This is the casino. At the far right of the casino, up the staircase and behind the table, lies a treasure chest containing the "Casino Passport".
Use the Casino Passport at EVS save points
The actual physical casino at the Durandal has very little to do with the Poker game, other than being the location of the passport and containing an EVS-enabled UMN Plate (save point). You can play Poker from any EVS-enabled save point in the game. To do so, enter the main menu, and select "Items". Use to scroll over to "Special Items", and select the Casino Passport. If you hear a buzzing sound, you are not properly positioned on the EVS-enabled save point. Exit the menu and try again when you hear a sound indicating you have touched the save point.
- EVS-enabled UMN Plates
- Onboard the Elsa, to the starboard side of the ship, just outside the bridge.
- The Durandal casino inside the residential area.
- The "Our Treasure" Inn in Sector 26 of the Kukai Foundation (except during the Gnosis attack).
How to play Poker
The common card game of Poker takes on many forms. The form used in this game is a simple one, known as "Five-Card Draw". Ordinarily a game of Poker is played between several players at a table, with each betting on the value of his hand. After each player has decided whether or not to invest in his hand, the hands are revealed and the player with the highest valued hand takes all of the money that had been bet. This game uses a slightly different version, to allow play by a single player. It is a form commonly used in American casinos, known as "Video Poker". In Video Poker, there are no opponents, and the goal is simply to make your hand as highly valued as possible. Depending on the value of your hand, you will either lose your initial bet or be returned some multiple of it.
The deck of cards
A Poker hand consists of five cards taken from a standard 52-card deck. The deck contains 13 denominations of cards in four different suits – the shape of the icon appearing on the card.
- Denominations of a standard deck
The Ace is normally considered either the highest or the lowest denomination, with the King being the next highest and the 2 being the lowest. The King, Queen, and Jack appear with pictures on them, while the Ace and the numbered 2 through 10 cards appear with a number of suit symbols equal to their denomination, with an Ace considered 1.
- Suits in a standard deck
Clubs are represented by a clover symbol. Diamonds are represented by a geometrical diamond. Hearts are represented by a heart, and Spades by a pointed leaf-shaped object. Clubs and Spades are black, and Diamonds and Hearts are red.
Xenosaga does not use a standard deck. In Xenosaga, the Ace does not exist, and is a 1 card. As such, it is always considered the lowest card, and the King is then the highest card. For the purpose of this page, the Ace will be used as the 1 card.
The value of a Poker hand
( Bet × x )
|Three of a Kind||3|
|Four of a Kind||20|
A Poker hand is more highly valued based on the presence of less probable card patterns. For example, having five cards of separate denominations and suits is the most common result, and thus the least valuable. Having five cards all of the same suit is significantly rarer, and thus is a highly valued hand.
The hand ranks are as follows:
- No Pairs
A hand consisting of five cards of separate denominations, without being all in sequence or all of the same suit. In ordinary poker, the highest card is considered the "value" of the hand for purposes of comparing with the other players. Thus:
would be considered worth an 8. If another player had:
the other player would win due to having a hand valued at 9. In the case of a tie, the second-highest card is compared. Such a hand is considered "9-high". In Xenosaga, such a hand is always a loser, and returns nothing.
- One Pair
A hand consisting of two cards of the same denomination, and no other pattern of note. Such a hand beats a hand without a pair, but loses to any other patterned hand. In the case of two hands with One Pair, the higher-valued denomination wins. In Xenosaga, any pair is valued the same, and returns the bet.
- Two Pairs
A hand consisting of four cards of two denominations, with no other pattern. Such a hand beats a single pair, but loses to any other patterned hand. Again, the highest valued pair is used as a tiebreaker if another hand also has Two Pairs. In Xenosaga, two of any pair is valued at two times the bet.
- Three of a Kind
A hand consisting of three cards of the same denomination, and no other pattern of note. Such a hand beats a hand with only pairs, but loses to other patterned hands. In Xenosaga, three of a kind is valued at three times the bet.
A hand consisting of five cards in numerical sequence, with no other pattern. For instance:
would be considered a straight. A straight cannot "wrap", so a King is always the highest card in a straight and cannot be the bottom:
The highest card in a straight breaks any ties, and a straight beats any paired or three of a kind hand. In Xenosaga, all straights pay out five times the bet.
This hand type is a bit different than the others, in that the denomination of the cards aren't considered (unless there is a straight). A flush is a hand with five cards all of the same suit. Ties are broken by the highest denomination card. Flushes beat straights, pairs, and threes of a kind. In Xenosaga, a flush is worth seven times the initial bet.
- Full House
A hand consisting of both a three of a kind and a pair. Such a hand beats anything except four of a kind or a straight flush. It pays out ten times the initial bet.
- Four of a Kind
Four cards of the same denomination constitutes a Four of a Kind. Such a hand beats anything except a straight flush. It pays out twenty times the initial bet.
- Straight Flush
A straight flush is a hand with both a straight and a flush, with five cards in sequence all of the same suit. It is the highest possible hand and beats anything, with ties broken by the highest valued denomination.
- Royal Flush
A royal flush is just a certain straight flush – one consisting of the highest cards available, all of the same suit:
Normally it would be Ace through 10, but Xenosaga doesn't have an Ace. While technically it is not a different hand type, since it cannot be beaten (only tied), it receives special payout in Xenosaga, of 100 times the bet.
Playing your hand
In Five-card Draw Poker, you are dealt five cards to start with. If you have one of the patterned hand types listed above, you are welcome to keep it and receive your prize. However, you usually don't get more than just a pair (if anything) on your initial deal. You have one opportunity to improve your hand by discarding unwanted cards and replacing them with new cards dealt at random. After this second deal, known as the "Draw", you are stuck with whatever you are left with. As such, it is usually a good idea to hang on to cards that form a pattern and discard other cards in an attempt to improve your pattern.
- Hang on to pairs
Holding this hand:
you would keep the two 2s and discard the rest, hoping to draw another 2 to get a three of a kind, or perhaps another pair for two pairs. If you're really lucky, you might draw two more 2s or even a brand new three of a kind for a full house.
- High cards are irrelevant
In Xenosaga, you are not competing against other players, but rather attempting to acquire as valuable a pattern as possible. As such, you don't have to worry about breaking ties, and a pair of 1s is just as valuable as a pair of Kings. Don't hang on to high cards in an attempt to match them up; pitch them to allow for more chances to improve your pairs.
- When to go for a straight or flush
If you are not dealt any pairs, you often want to throw away the entire hand and get a new one, since you have nothing you want to work with. However, sometimes you will be dealt no pairs, but several cards that look like they might make up a superior pattern, such as a straight or a flush. In that case, you can choose to hang on to the partially completed pattern in an attempt to finish it off. For example, holding this hand:
you might discard the 8 in an attempt to draw either a 1 or a 6. Either card would complete a straight. However, the odds are against you, in that only two denominations will complete your straight, while eleven will not. Still, four others will get you at least one pair, so it's not all that bad a deal. Avoid drawing to inside straights, however, such as this hand:
since only one card is capable of completing the straight, the 4. Similarly, straights that are blocked off by an extreme card, such as this hand:
have only one card that will complete them, and are thus bad news. It's better to discard the entire jumble and hope for at least a pair then to hold on to a faint hope at a better pattern.
Flushes pay out extremely highly, but are relatively rare. However, sometimes you are dealt four cards of the same suit, and one card of a different suit. In that case, you can choose to pitch only the extraneous card and holding on to the almost-flush. Still, be aware that although it might appear you have 25% odds to complete the flush, it's actually a bit lower than that, since four cards of that suit have already been removed from the deck. Assuming your initial looks like this:
there are only nine Diamonds and 38 other cards left in the deck, for about 19.1%, a bit lower than 25%. If your fifth card makes a pair with a card in the almost-flush, it's usually not a good idea to break up that pair in a futile attempt at a flush, when you could go for a three of a kind or two pair instead.
- High value hands
Very highly valued hands such as Straight Flushes are extremely rare, and generally occur more by fluke than by actually attempting to complete them. Still, holding a hand like this:
it is very, very tempting to take that tiny chance at getting the missing 9 and 10. It's probably not a very good move from a probability standpoint, but it can be fun as long as it's not ridiculous, such as trying for a royal flush holding two of the required cards, or done regularly with only three. Still, in general you should go for the common patterns, especially due to the presence of the High & Low game.
Poker in Xenosaga
In Xenosaga, you can't directly gamble away G, the currency of the Xenosaga universe. Instead, you must buy "Coins", which can be gambled at either the Slot machine or the Poker game. Coins can be used to buy a variety of "prizes", which will be detailed below. To buy coins, access the Casino using the Casino Passport, and select "Exchange". You can then select "Purchase Coins" from the menu.
Coins are available in the following packages:
- 10 Coins: 100 G
- 100 Coins: 950 G
- 500 Coins: 4,500 G
- 1,000 Coins: 8,000 G
While it might seem like a better deal to purchase the Coins in bulk, it really is a waste of G. These are horrifically high prices for Coins, which can easily be acquired by simply winning the Poker game. If you have a bit of money to spare, you can start with 100 Coins so you don't have to worry about coming back for more, but if you want to be cost-efficient, you can buy just 10 and start at the low-stakes machines. You may have to buy a few batches of 10 before you win, but once you hit a ×16 payout on the High & Low game, you will not need to buy another Coin ever again. Still, you will soon be racking in a virtually unlimited amount of cash, so if you want to get started immediately on the high-stakes (if you call 100 Coins high) game, feel free to buy a larger package.
The Poker game
Heading back to the main Casino menu, once you have Coins, select "Poker" to begin the Poker game. The Poker game can be played in four different levels of stakes:
- LEVEL-1: 5 Coins
- LEVEL-2: 10 Coins
- LEVEL-3: 30 Coins
- LEVEL-4: 100 Coins
Since the Poker game is biased heavily in your favor, you will want to play for as high of stakes as you can afford. However, if you initially purchased just a few Coins, you might want to start out at a lower level, such that you don't run out of coins and have to buy more.
- About probability
You may be wondering, if this is such as sure way of gaining money, how can you run out of coins? Consider the saying, "The House always wins". This is an old axiom about casinos – they turn a profit. In order to do such, they have to be making money on the gambling taking place within. Yet, it is still possible to show up at a casino and go home a winner. How is this possible? Statistical sample size is the answer. In a casino, the games are set up such that the probabilities favor the house ever so slightly. Thus, any game can be won or lost by anyone, and five or ten or even fifty games can go either way, but over the long haul the casino will make money. This is the principle of the law of averages.
In Xenosaga, the probabilities favor you. And it's not just a slight favoring, it's hugely, immensely in your favor. And yet, after two or three or even ten hands, it's possible you might lose some money. Due to the overwhelming odds, the law of averages will kick in pretty soon after that, and you'll be sure to turn a profit over even a short time. Still, a couple hands here and there can go against you. Once you have about 1,000 Coins, you will never have to worry about running out again, and you should get there very quickly.
Over time, you will make a ton of money with the Xenosaga Poker game.
- Game mechanics
Once you select a level, you will be presented with a screen detailing how many Coins you have, what the payout levels are, and a dialog box asking you if you want to play that level of Poker. Select "Yes", and you will be given your hand.
Below each card is a "DRAW" button, with a separate draw button in the middle of the screen. The "DRAW" buttons below the cards are used to toggle whether or not you wish to keep each card. By selecting one, it changes to "HOLD". Now the card will not be pitched when you go to make your draw. If you mistakenly choose to HOLD a card, you can select the "HOLD" button to toggle it back to "DRAW". When you eventually select the main "DRAW" button in the center, all the cards that are marked "DRAW" will be jettisoned, and new cards will be dealt in their place.
After the second deal, your hand will be evaluated, and, if you have at least a Pair, the value of your hand will light up on the chart. If you do not have at least a Pair, you have lost, and will be given the option to play again.
If you won at least a 1× payout, you will be given the option to play a "Double or Nothing" game, entitled "High & Low". By playing "High & Low", you can multiply your payout by anywhere from two to sixteen times – or you can lose it all.
High & Low
When you win at least a 1× payout in the Poker portion of the Poker game, you will be given the option to play "High & Low". If you choose to do so, you will enter a different screen, in which five card slots appear at the top of the screen, and a set of indicators from 2× – 16× appear where the payouts normally are.
This game is very similar to the old television game show "Card Sharks". In this game, you will be presented with a card facing you. You are given the opportunity to guess whether the next card will be of a higher or lower denomination than the current card. You also can choose to stop at any time, even after seeing the card. If you choose to go on, you will be dealt another card. If it fits the guess you made, you will double your payout, and if not, you will have lost it all. When the same denomination is drawn, you win regardless of your guess. You can continue this until you choose "Stop", lose, or reach a payout multiplier of 16× (four consecutive correct guesses).
Whatever the result, when you are done, you will be returned to the regular Poker mode, to start anew. You can play High & Low any time you earn any payout in the Poker mode.
This is the portion of the game where the real money is made. The next section is about playing High & Low and making huge amounts of money.
Making lots of money
Video poker alone isn't going to get you much. If you never go for double or nothing, you'll settle around the amount of money you started with, occasionally winning a one or two times payout, and often losing. Every once in a while, you'll get a ten times payout or more, and if you're ridiculously lucky, you might get 100 times back from a Royal Flush once in a blue moon.
Still, even 100× is only 10,000 Coins, which isn't going to get you anywhere. Wouldn't it be nice if there was a way to routinely rake in huge amounts of Coins, such as 1,600 from a simple pair, or 3,200 from two pair? Imagine getting 4,800 from a three of a kind! That's half as much as a Royal Flush, and it comes up thousands and thousands of times more often. Still, you are only allowed to gamble 100 Coins at a time, and even if you were able to bet more, you'd have no guarantees of actually winning consistently over time at the simple Poker game.
The solution to all of this is to use the "High & Low" game, which is ridiculously balanced to favor you – at an expected rate of payout of 5.5 times what you put in! That means that over time, your pairs will be worth 550 each – and you certainly get enough pairs to make that worthwhile.
Playing the "High & Low" game feels a bit dangerous, especially when the amounts get big, and the card isn't a nice friendly one such as a Queen or a 3, where it is extremely likely that the direction you pick will turn up. Still, in order to secure really fast, effective, and consistent payouts, you must risk it all.
That's right, even on a full house already gone to 8x and a 7 showing, you must go on. The odds are in your favor every time, even with the worst possible card faced, the dreaded 7. It might feel frustrating to lose an 8× full house, but for every one you lose, there will be even more 16× payouts that you would have otherwise missed. Remember, the odds are in your favor – always, every time, no matter what.
Here's the table of odds, assuming you pick logically, meaning LOW on 8 through King and HIGH on 1 through 6 and whatever you like on 7 – so long as you don't stop!
- King: 100.00%
- Queen: 92.31%
- Jack: 84.62%
- 10: 76.92%
- 9: 69.23%
- 8: 61.54%
- 7: 53.85% (Yes, even this is in your favor over the long haul, which is what we're playing for.)
- 6: 61.54%
- 5: 69.23%
- 4: 76.92%
- 3: 84.62%
- 2: 92.31%
- 1: 100.00%
If you do not go on every time, all you're doing is slowing down your gains. It's irrelevant if you blow this 1,600 – you're playing for hundreds of thousands, not a few measly Coins here and there. In fact, as the calculations below show, it turns out that if you always go on, you will get a 16× payout about a third of the time. The other two thirds you will lose it all. That's 5.5× on average, which is a ridiculous expected payout for a gambling machine. No real gambling machine ever pays out above 1× on average, it would be suicide for the casino. 0.95× is a great payout. This one pays out 5.5! It would be a steal at 1.1, but now it's just ridiculous.
You can make about 200,000 Coins per hour if you follow this simple system, and that's just with pairs and threes of a kind. In order to make 200,000 Coins without "High & Low", you'd need to score twenty Royal Flushes without losing. And even one Royal Flush is so unlikely as to be irrelevant. But – if you get one, remember – keep going, even on 7!
The system never fails; you'd have to lose 16 times in a row between successes just to break even on one single pair paid out through a 16×. That doesn't happen often; far more often you pull through another 1,600 from another pair, and then a 4,800 from a three of a kind. Yes, it's frustrating to lose 8,000 from an 8× full house, but you'll get that 8,000 right back in two minutes – far better to take the ODDS-ON bet to get to 16,000. And more often than not, those 8,000 full houses will become 16,000 full houses. Even when a 7 is showing.
You're playing for the long haul, not the short term. As such, it is your goal to maximize expected payouts, just like a casino does. A casino doesn't mind the occasional player who hits the jackpot, since it's a certainty that for each jackpot, there are numerous losses going right into their pockets. And here, you get to be the casino – you get to experience what it's like to have the odds in your favor.
If you still aren't convinced, read the section on the mathematics behind "High & Low", or just follow the system for 15 minutes. You'll see in no time that it works.
So you've made a lot of Coins, probably hundreds of thousands, playing Poker and High & Low. Now, the question remains – how do you get cold hard G out of it? You can only spend Coins on selected items at the casino prize store, none of which are the famed AG-05 AGWS.
Cashing out proves to be an extremely tedious process. The quickest way to do it is to go to the EVS save point on board the Elsa, which is directly next to a UMN Silver Plate, where you can sell items for G. Empty your inventory of Med Kits, Ether Packs, Revives, and Cure-Alls, and go back to the Casino using the Casino Passport. Select "Exchange", and this time select "Prize Exchange". You'll be presented with the items in the table below. The first item, the Recovery Set, will be your source for G. Claim 99 Recovery Sets, for 9,900 Coins by hitting 99 times. The easiest way to do this is not to count, but to determine your finishing point, e.g. if you have 328,740 Coins, you will jam on until you are down to 31,9740 – 9,900 less than you started with. Then, exit the casino and go back to the Silver Plate. Sell off all your Med Kits, Ether Packs, Revives, and Cure-Alls again, which should net you 990 for the Med Kits, 1,980 for the Ether Packs, 2,970 for the Revives, and 4,950 for the Cure-Alls. The total for all of that comes to 10,890 G, for and exchange rate of 9,900 Coins to 10,890 G, or nearly 1:1.
This takes time, though, since you constantly have to reload the Elsa, then the shop, then the Elsa, then the menu, then the Casino, then the Elsa, not to mention all the button jamming. Overall, it takes about one full minute to transfer one set of 99 Recovery Sets, or about one minute per 10,000 G. That's 300,000 G, enough for the AG-05, transferred out in about half an hour. Still, it's tedious work, much less exciting than playing Poker and High & Low.
If you're interested in Xenocard, you can cash out booster packs extremely cheaply, at only 100 Coins each, and you don't even have to go through the selling. With hundreds and thousands of booster packs, you will easily get every (non-promotional) card in the game, even rares, in sets of three, allowing you to make any deck you want. Later in the game, you can even get several promotional cards from the Casino. Finally, you can check out some nice production sketches for next to nothing, considering how quickly you can acquire Coins.
Just remember to buy a full set of Recovery Sets after you finish selling them, so that you don't find yourself in a dungeon without Revives or Cure-Alls that might be essential.
The list of prizes
The one time only Xenocard packs are only available after completing the Song of Nephilim.
|Cost||Name||What it does||Availability|
|100||Recovery Set||1x Med Kit, Ether Pack, Revive, Cure-All||Unlimited|
|150||Escape and Rest Set||1x Escape Pack, Bio Sphere||Unlimited|
|10,000||Golden Dice||Access- Fluctuating damage based on HP||One time only|
|15,000||Bravesoul||Access- Strength+ when HP low||One time only|
|18,000||Revive DX||Item- Revives with max HP||One time only|
|12,000||Stim DX||Item- PATK+50% for one fight||One time only|
|2,000||Design Sketch 01||Shion 1||One time only|
|2,000||Design Sketch 02||Shion 2||One time only|
|2,000||Design Sketch 03||Shion 3||One time only|
|2,000||Design Sketch 04||chaos 1||One time only|
|2,000||Design Sketch 05||chaos 2||One time only|
|2,000||Design Sketch 06||chaos 3||One time only|
|2,000||Design Sketch 07||Jr. 1||One time only|
|2,000||Design Sketch 08||Jr. 2||One time only|
|2,000||Design Sketch 09||Jr. 3||One time only|
|2,000||Design Sketch 10||Jr. 4||One time only|
|2,000||Design Sketch 11||MOMO 1||One time only|
|2,000||Design Sketch 12||MOMO 2||One time only|
|2,000||Design Sketch 13||MOMO 3||One time only|
|2,000||Design Sketch 14||KOS-MOS 1||One time only|
|2,000||Design Sketch 15||KOS-MOS 2||One time only|
|2,000||Design Sketch 16||Ziggy 1||One time only|
|2,000||Design Sketch 17||Ziggy 2||One time only|
|2,000||Design Sketch 18||Ziggy 3||One time only|
|2,000||Design Sketch 19||Gaignun 1||One time only|
|2,000||Design Sketch 20||Gaignun 2||One time only|
|2,000||Design Sketch 21||Elsa||One time only|
|2,000||Design Sketch 22||AG-01||One time only|
|2,000||Design Sketch 23||Cockpit||One time only|
|2,000||Design Sketch 24||VX-9000||One time only|
|2,000||Design Sketch 25||AG-04||One time only|
|2,000||Design Sketch 26||VX-20000||One time only|
|2,000||Design Sketch 27||VX-4000||One time only|
|2,000||Design Sketch 28||AG-05||One time only|
|2,000||Design Sketch 29||Shion CG||One time only|
|2,000||Design Sketch 30||KOS-MOS CG||One time only|
|400||Starter Set||Xenocard- Starter Deck||Unlimited|
|100||Card Pack #1||Xenocard- Booster Pack 1||Unlimited|
|100||Card Pack #2||Xenocard- Booster Pack 2||Unlimited|
|1,000||PM Card F||Xenocard- AG-05 Promotional Cards||One time only|
|1,000||PM Card G||Xenocard- Third Armament Promotional Cards||One time only|
|1,000||PM Card H||Xenocard- Testament Promotional Cards||One time only|
|1,000||PM Card I||Xenocard- AG-04 Promotional Cards||One time only|
|1,000||PM Card J||Xenocard- Phase Transition Cannon Pro. Cds.||One time only|
|1,000||PM Card K||Xenocard- Invoke Promotional Cards||One time only|
|1,000||PM Card L||Xenocard- Destiny Promotional Cards||One time only|
|1,000||PM Card M||Xenocard- Dammerung Promotional Cards||One time only|
|1,000||PM Card N||Xenocard- So Weak! Promotional Cards||One time only|
|1,000||PM Card O||Xenocard- Rhine Maiden Promotional Cards||One time only|
|1,000||PM Card P||Xenocard- Unknown Armament Promotional Cds.||One time only|
|1,000||PM Card Q||Xenocard- Proto Dora Promotional Cards||One time only|
The mathematical evidence
At last, the heavy part of this guide. The claim has been made that the "High & Low" machine is hugely weighted in favor of the player, with an expected payout of about 5.5 times what you put in to it. This fact has been used to argue that you should always play on until you get the 16× multiplier, regardless of the 7s, 8s and 9s along your way. Presented here is some evidence of this, based on simple mathematics.
How can you calculate the expected payout of something as complicated as a series of decisions like this? The answer is that there actually is no decition at all taking place in the "High & Low" game! A winning layout is always a winning layout, and a losing layout always loses, assuming the player chooses logically, according to probability, whether the next card will be high or low. For example, consider this layout:
Assuming the player doesn't go against the odds, this will always win! It's like a lottery scratch-off game, in that the results are predetermined, and you are simply slowly revealing whether you have a winner or loser. A sane player will always pick "Low" on the King, "High" on the 6, "Low" on the 9, and "High" on the 4. Thus, the logical player always wins with this layout.
Similarly, this layout should always lose. There is no reason a sane player would ever pick "High" on the 10, and thus the player will always lose to the Jack. Because of this, we can analyze all of the possible layouts and determine how many winners there are and how many losers.
This is a slight oversimplification, however. The truth is, that a 7 card presents a dilemma. Either "High" or "Low" present equal probabilities of winning. Thus, a layout like:
might be a winner, if the player picks "Low", or it might be a loser if the player picks "High". Thankfully, this does not present a real problem from a mathematical sense. Regardless of which option the player picks on the 7, there are an equal number of winning and losing layouts. We can simplify the mathematics by assuming the player always picks "High" on a 7, but the math will work out the same regardless of what system you use to pick your 7s.
Sampling – with or without replacement?
There is one more simplification necessary, in order to make the math immensely less difficult. However, this simplification, unlike the 7 issue, actually does slightly affect the results.
When a card is selected from a deck of cards, it is removed from that deck and placed face up on the table. If the King of Spades is picked, there is no longer a King of Spades left in the deck, and so it cannot be picked again. This concept is known as "Sampling without replacement". This makes any mathematical analysis of the problem ridiculously complicated, since every card selected modifies the probabilities of every other card in the deck. For instance, when the King is showing as the first card, the probability of the second card being a King is only 3/51, while any other card has a probability of 4/51. This is because there are only three Kings left in the deck, and four of every other card.
A model can be created, however, where the card that is selected is still available in the deck to be selected. This will not give you an exact mathematically sound analysis of the game, but it will provide you with an extremely close approximation. Since the whole point of this section is to show that the "High & Low" game is weighted heavily in your favor, and not to determine the exact probability to the nth decimal place, this model, "Sampling with replacement", is used to make the math bearable. Now there is always a 1/13 chance of drawing a King, regardless of which cards are showing.
This simplification will provide you with a valid approximation as to the expected payout of the "High & Low" machine without requiring a degree in statistics.
In order to determine the probability of a winning layout, two things need to be counted: the total number of layouts and the number of winning layouts.
Counting the total number of layouts in our "Sampling with replacement" model is easy: there are five slots, which can contain any of 13 cards each. Thus, there are 135 total possible layouts. (Some of these layouts aren't really valid, such as five 2s, but these are very few and are a result of our model being used instead of the actual game without replacement.)
- Total layouts
Now, all that's left is to count the winning layouts. This is a bit more difficult, since a winning layout isn't readily visible through simple mathematical methods. However, something else that will give the same result in the end, would be to count the number of losing layouts. This is much easier to do, since the number of layouts that start with a losing combination, such as 10-Jack, can be determined and those subtracted off.
This is done in four separate steps, since a losing layout can occur at any of the four decision points. However, once a layout is a loser, there is no point in checking it again – it has already lost. So, starting at the beginning, you need to find out how many layouts are losers after the first round. Then, the number of remaining layouts can be counted, and check only those to see if they lose in further rounds.
Subtract losing layouts
There are four decisions that need to be made successfully in order to pay out. This will determine the chance of surviving all four decisions. You can note from the tables below that even with the worst card, a 7, showing, there is still a greater than 50% chance of surviving the round.
To explain the following tables:
- The "Card" column indicates the faced card (represented by the joker).
- The "#L" and "#W" columns indicate the number of different cards that will lose or win when flipped (represented by the red-backed card).
- The "#L Layouts" and "#W Layouts" columns indicate the total number of layouts that will win or lose this round, given that initially faced card. This is determined by multiplying the number of losers by 133, and the same for the losers, to represent any card in the blank slots to be revealed later (represented by blue-backed cards).
- The "Total Losers" column keeps track of a running total of losing layouts with each card faced.
- The "Win%" column shows the approximate chance of winning this round given the faced card. This is always greater than 50%, even with a 7!
- Round 1
|Card||#L||#W||#L Layouts||#W Layouts||Total Losers||Win%|
That leaves 79,092 layouts that lose on the first of four decisions.
- Result: 292,201 winning layouts, for a 78.70% chance of surviving Round 1.
- Round 2
Now, to examine the possible cards that are left for Round 2. What was advanced with? First, analyze all possible winning combinations from the previous round:
|If the faced card is a...||... you can advance with any of these.|
Now, add up the number of occurrences of each card to determine the frequency of this card being used to start the next round. As you can see from the table, 8 and 7 are a bit more likely to be showing up here than numbers at the extreme ends.
- King: 8
- Queen: 9
- Jack: 10
- 10: 11
- 9: 12
- 8: 13
- 7: 13
- 6: 12
- 5: 11
- 4: 10
- 3: 9
- 2: 8
- 1: 7
This time, there are only three cards left to flip. We can calculate the number of losing layouts with each card showing, but then we have to multiply by the number of cases in which this card will be showing to get the total number of times this combination occurs.
The new "xOccur" column represents this. Also, the "Total Losers" column is now multiplied by "xOccur", to count all of the losers regardless of what the first card is.
|Card||#L||#W||#L Layouts||#W Layouts||xOccur||Total Losers||Win%|
That leaves 69,797 layouts that lose on the second of four decisions. There were 292,201 winning layouts from the first round, giving us a chance of 76.11% of surviving specifically Round 2, and a 59.90% chance of making it all the way to round 3.
- Result: 222,404 winning layouts, for 59.90% chance of surviving Round 2.
- Round 3
For this round, the number of combinations that will start with each specific card must be determined again. This time, however, it's not as simple, since the third card likelihood is derived from the second card likelihood (remember that certain cards are less likely to be showing since we tend to lose with them, ending the game).
How often is a King showing for the third card? A King shows up eight times out of thirteen as a winner, and the other five times it ends your game. However, every time a 7 shows up, it wins – there is no case where a 7 is a loser if you always pick according to the odds. Because of this, you're a lot more likely to be seeing 7s at this point than extreme numbers, since often the extreme numbers ended your game.
Counting how many combinations there are that leave you with each card, is not so easy this time. The number of combinations from Round 2 for cards that would make the current card a winner are added up to figure out the total number of combinations that would leave you with this card faced.
|Card||Wins on||Adding...||...to get.|
This time, the math isn't so obvious, so to verify that this actually works, this is a list af all 78 non-losing combinations that end with a king faced:
These combinations can be listed out for any other card as well.
The tables for Round 3 can be made, since the number of appearances for each card are now known.
|Card||#L||#W||#L Layouts||#W Layouts||xOccur||Total Losers||Win%|
That leaves 53,599 layouts that lose on the third of four decisions. We had 222,404 winning layouts from the second round, giving us a chance of 75.90% of surviving specifically Round 3, and a 45.46% chance of making it all the way to round 4, the final round.
- Result: 168,805 winning layouts, for 45.46% chance of surviving Round 3.
- Round 4
To calculate the number of combinations with each card faced for the final round, we can use the same method we used for Round 3, plugging in the Round 3 numbers in place of the round 2 numbers.
So, for example, the King calculation would look like:
|Card||Wins on||Adding...||...to get.|
Now the last round losers can finally be trimmed off, leaving the number of winning combinations. (The layouts that lost before the last round have already trimmed off.)
|Card||#L||#W||#L Layouts||#W Layouts||xOccur||Total Losers||Win%|
That leaves 40,684 layouts that lose on the fourth of four decisions. There were 168,805 winning layouts from the third round, giving the chance of 75.90% of surviving specifically Round 4, and a 34.51% chance of making it all the way through all four rounds and coming out a winner.
- Result: 128,121 winning layouts, for 34.51% chance of paying out.
Assuming the sampling simplification is a reasonable approximation for the actual probabilities of the "High & Low" portion of the Poker game, you can see that the approximate chances of paying out 16× is 34.51%. That's very near a third of the time. So, assuming you always play "High & Low" on every winning Poker hand, and always continue on to the end regardless of what numbers show up, you will multiply your poker winnings by 16 approximately one third of the time, and go home empty two thirds of the time. Using simple probability, the expected multiplied payout of "High & Low" is:
That means over the long run you will win 5.5 times what you put in. This is a huge bias in favor of you, the player. In a real-life casino, you will be lucky to get a machine that pays out a bit below 1. Any casino that paid out any multiplier over 1 would go broke in no time, and if a casino could pay out the inverse of the "High & Low" game, (1/5.5216) or 0.1811, they would rack up fortunes as fast as you can! Of course, nobody in their right mind would play on a machine such as that. Thankfully for you, the Durandal casino isn't in its right mind!