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Kniffel chance

kniffel chance

Bei Kniffel wird die untere Tabelle mit dem Feld „Chance“ komplettiert. Ein Spieler, der keines der erforderlichen Ergebnisse gewürfelt hat, kann, an Stelle einer. 6er. Summe oben, 0. Bonus (bei 63+), 0. Summe oben, 0. Dreierpasch. Viererpasch. Full-House. Kleine Straße. Große Straße. Kniffel. Chance. Summe unten, 0. Jan. Kniffel ist ein beliebtes Würfelspiel für die ganze Familie. Sie brauchen nur fünf Chance - hier muss nichts zusammenpassen. Alle Augen.

chance kniffel -

So kann es auch mal sein das z. Einer bis Sechser mindestens einmal die jeweilige Augenzahl: Daraus ergibt sich eine mittlere Punktzahl von 10, Bei einem doppelt mittleren 3er gibt es 2 Würfel mit den Augenzahlen 2 und 5 und 1 Würfel mit der Augenzahl 3 oder 4. Die mittlere Punktzahl für eine Chance beträgt bei optimaler Strategie 23, Zum Verständnis der Kategorien betrachtet man von jeder vorhandenen Augenzahl jeweils nur einen Würfel. Bei fünf Einlingen wird nur ein Einling behalten, egal welcher. Es gewinnt der Spieler , der am Ende am meisten Punkte hat.

You can play Yahtzee Club alone, with another player online or against the computer AI. No Flash, no Java needed or no install needed.

You can play it in any Browser your PC or mobile. If you play with the computer robot artificial intelligence, it uses the best strategy, so to win you need some skill and luck.

You can also ask for advice from the computer, so it can teach you how to win. High Score list is maintained on a Game Server.

More information related to Yahtzee can be found Wikipedia. Server application of the game is Java application running in the Google App Engine.

Rules The aim of the game is to collect maximum amount of points by throwing five dice. You can ask for advice or hints from the computer AI, so the robot can teach you how to win Scoring You can score in any of the scoring buttons that are not already selected.

You can select an individual game from the window to see the exact points in each combination Network Scores Best results are saved on the game server.

You can browse the list selecting Network Scores You can select an individual game from the window to see the exact results in each combination If a result from your computers Score list is better than it will be also sent to server Multiplayer NetworkGame If you want to play in network Login first, then choose "Network".

You can wait until someone invites you into a game or you can send an invitation to a player selected from the list "Select Opponent" Preferences In preferences you can set the user name you want to use in Network Game.

Cookie Policy This website uses cookies for statistics, advertising, game scores and more. By using this website you accept its use of cookies.

Kniffel oder Yahtzee ist ein Würfelspiel mit fünf Würfeln , einem Würfelbecher und einem speziellen Spielblock. Das Spiel ist kommerziell erhältlich, wird jedoch häufig ohne den vorgedruckten Block gespielt.

Es leitet sich ab vom Escalero , einer südamerikanischen Form des Würfelpokers , und dem Yacht , ist jedoch wesentlich jünger und wird mit herkömmlichen Spielwürfeln gespielt.

Hinzu kommen verschiedene Varianten und Umsetzungen, etwa als Reisespiel oder als Computerspiel. Die ursprünglichere Form, das Yahtzee, wurde von E.

Lowe in den Vereinigten Staaten auf den Markt gebracht. Später erschienen weitere Varianten mit geringen Unterschieden. Yams sowie in den Vereinigten Staaten als Yatzy auf den Markt.

Dabei unterscheidet sich Yatzy von Yahtzee und Kniffel darin, dass als zusätzliche Wertungen der einfache und der zweifache Pasch aufgenommen wurden.

Yatzy wird auch in der Schweiz und teilweise in Österreich gespielt. Im englischsprachigen Raum ist zudem die Version Kismet bekannt. In Dänemark und Teilen von Südostasien ist das in den Grundzügen ähnliche Balut populär, das sich in der Wertung jedoch deutlich von Yahtzee unterscheidet und direkt aus einem Pokerwürfelspiel hervorgegangen ist.

Hinzu kam der Bonus im oberen Block, der vergeben wird, wenn bei allen Zahlenwürfen mindestens je drei Mal die gleiche Würfelzahl geworfen wird.

Firstly, the Optimal strategy takes no account of any opponents. In normal gameplay a player will adjust their strategy depending on the scores of the other player or players.

Secondly, the Optimal strategy tends to give undue importance to Yahtzee bonuses. It is rare for a player without a Yahtzee bonus to beat a player with one.

So, in normal gameplay, a players' strategy is not significantly influenced by the value given for a Yahtzee bonus.

Consider the situation where the Yahtzee bonus was worth a million rather than a hundred. It would not influence normal gameplay where the objective is to score more than the opponent.

It would, however, affect the "Optimal" strategy since scoring a million would have a dramatic effect on the average score. The "Optimal" strategy would be dominated by the prospect of a Yahtzee bonus.

Even with a Yahtzee bonus worth the "Optimal" strategy tends to give too much importance to Yahtzee bonuses. Despite these limitations the "Optimal" strategy does provide a useful guide as to the best strategy, especially in the early rounds.

The "Optimal" strategy for the first round is described in the next section. After the first round, the number of different games rapidly makes detailed analysis difficult but, in the early rounds, players generally simply adapt the first round strategy based on the boxes used.

For instance if a player throws in the first round they will keep 66 but if the 6s box was used in the first round and they throw in the second round they will naturally keep 55 instead.

The following table shows the average score obtained using the Optimal strategy and the proportion of the time that zero is scored in a particular category: It is possible to calculate the maximum average score with different rules.

When the rules are changed so that there is no Upper Section bonus the average score drops from This compares with an average bonus score of The average number of Yahtzees thrown in a game is 0.

In about a quarter of these cases the Yahtzee box has a score 0 and so there is no Yahtzee bonus. Although the average score is The 1st Percentile is i.

The strategy for the first round described here is based on that used by the "Optimal" strategy. Sometimes there are two possible plays which are almost equally good.

In these cases the simplest option is given, i. Although there are different ways of throwing 5 dice, because the order of the dice is not important there are only different combinations.

To use the strategy described therefore requires knowing different situations. The general principle after the first and second throws is to keep the largest number of similar dice and rethrow the rest.

When a player has a full house he will keep the three-of-a-kind. With two pairs, keep the higher pair and rethrow the other three dice.

If a large straight is rolled, keep it. If a small straight is rolled, keep it and re-roll the fifth die. If all the dice are different and there is no straight, keep only the 5.

There are a number of exceptions to these rules. These exceptions are different for the second and third rolls. So, before the second roll he will never keep two pairs, he will never keep a pair of 1s and he will never throw all the dice again.

So there are specific differences compared to the second roll. Now he will keep the Full House if the three-of-a-kind is 2 or 3 and keep two pairs when they are 1 and 2 or 1 and 3.

He will Keep all large and small straights. He will Normally keep a pair of 1s although is still better and keep rather than 5 when all the dice are different.

If he has any of the following he will put it in the appropriate Lower Section box: Yahtzee, full house, large straight, small straight.

With a 4-of-a-kind or 3-of-a-kind he will put it in the appropriate Upper Section box. The only exception is that when he has a 3-of-a-kind where the sum of the dice is 25 or more he should use the 3-of-a-kind box.

The 4-of-a-kind box is never used in the first round. In these situations there is no appealing option, the choice being between using an Upper Section box and using Chance.

With two pairs follow the rule for the lower of the two pairs he has. The rule depends on whether his pair or smaller pair is low 1, 2 or 3 or high 4, 5 or The best outcome in the first round is to throw a Yahtzee and put it in the Yahtzee box.

This increases the final expected score from The second best outcome is to score 24 in the 6s box. This has a value of The worst first roll is which has a value of The worst third roll is which has a value of The highest possible score is 1, and involves throwing 13 Yahtzees, scoring 12 Yahtzee bonuses of points each, as well as in the other categories.

The is achieved by scoring five aces 5 , five twos 10 , five threes 15 , five fours 20 , five fives 25 , five sixes 30 , the Upper Section bonus 35 , five sixes scored as Three-of-a-kind 30 , five sixes scored as a four-of-a-kind 30 , a full house 25 , a small straight 30 , a large straight 40 , a Yahtzee 50 , and five sixes scored as chance The score of 1, requires using the "Joker" rules for the full house, small straight, and large straight categories.

In order to score 1, the player must throw 13 successive Yahtzees. The probability of getting a Yahtzee, if that is the sole objective, is 4.

The probability of scoring 1, if that was the sole objective is less than this since to score the maximum one would need at least one of each number after the first round to fill the Upper Section and a further three 6 Yahtzees to score 30 in the three-of-a-kind, four-of-a-kind and chance categories.

This most frequently happens when a player rolls two pairs. This means that a player can slightly increase the chance of getting a 6 Yahtzee without reducing the chance of getting a Yahtzee.

In normal gameplay the probability of scoring 1, is considerably less. For a player following the strategy which maximises the chance of getting a Yahtzee, the probability of scoring a particular number of Yahtzees follows a binomial distribution and is shown in the following table.

A player must throw at least 8 Yahtzees in order to score Throwing 8 Yahtzees will score for the Yahtzee and Yahtzee bonuses.

The 7 Yahtzees after the first throw will generally score highly, including the opportunity of using the Joker rule to score in, for instance, the Large Straight category.

This means that a player throwing 8 Yahtzees will often be able to score 1, In normal gameplay the probability of scoring this number of Yahtzees is considerably less.

The following table shows the probability of getting certain scores using the "Optimal" strategy. The lowest possible score is 5. The Chance box always scores the sum of the dice, so that a minimum of 5 must be scored in that category.

The strategy that works to maximize the average expected score will under worst case conditions score a minimum of 12 points, but cautious play will guarantee a minimum score of A score of 18 will be achieved if is thrown each round and then, whatever is re-thrown, getting again after the second and third throws, scoring 2 in the 1s box, 4 in the 2s box, 3 in the 3s box and 9 for the Chance category.

Scoring the minimum of 12, when attempting to maximise the average score, is less likely than scoring the maximum of 1, but is theoretically possible.

Suppose that in the first two rounds after each roll. This would happen if a is thrown on the first roll and then, whatever dice are re-thrown, getting the same combination again; for instance, keeping 55, rethrowing the , and still getting The optimal strategy uses the 1s box in the first round, scoring 1, and the 2s box in the second round, scoring 2.

Now suppose that for the next 3 rounds results after each roll. Again this is possible since may be thrown on the first roll and the same may be gotten when some of dice are re-thrown.

The optimal strategy uses the four-of-a-kind and Yahtzees boxes in the third and fourth rounds and then the 3s box in the fifth round, scoring 0 each time.

For the last 8 rounds suppose is rolled each time. Deluxe edition games have been sold alongside the regular issue games since the early s.

They all contain components that are more luxurious than standard game parts. In recent years, a number of collector issue Yahtzees have been sold as well.

Some of these collector issues have dice that replace the pips with certain symbols connected to a theme, but still correspond to the numbers one to six.

Since the s, Travel Yahtzee has been sold in various forms as part of Milton Bradley's line of travel games. Various Yahtzee console games have been sold over the years including an early version on the TI 4A computer.

In , the game was first released to PC and Mac users by Atari.

It is important to understand the probability of completing a Yahtzee. This depends on whether a particular Yahtzee is required or any Yahtzee.

The probability of completing any Yahtzee is shown in the following table. The strategy is to keep any Yahtzee, four-of-a-kind, three-of-a-kind or pair that is thrown and re-roll the others.

With two pairs, either can be chosen. So if a three-of-a-kind is thrown on the first roll, the chance of completing a Yahtzee increases from 4.

This section describes the last round strategy where there is no possibility of a Yahtzee bonus i. These are the simplest situations to analyse, although even at this late stage the strategy may vary depending on the game situation.

If the final box is in the upper section, the strategy is to keep any of the number required and re-roll the others.

The distribution is as follows:. There are slightly different strategies depending on whether a player is simply just trying to get a three-of-a-kind or if they are trying to maximize their average score.

Different strategies will also be required should a specific target be needed to achieve. The strategy to maximize the chance of getting a three-of-a-kind involves keeping any three-of-a-kind that is rolled.

If a three-of-a-kind is rolled then after the first throw the player should keep any other 5s and 6s, while after the second throw the player should keep any other 4s, 5s, and 6s.

An example is with the player keeps and throws the other 2. If a three-of-a-kind is not rolled the player should keep any pair that is rolled and re-roll the other dice, with two pairs the player should keep the higher pair, and with no pair the player should keep the highest die.

Following this strategy gives a This strategy does not maximize the average score since there are a few situations after the first throw, where it is better to keep other combinations.

For instance after throwing , keeping maximizes the chance of getting a three-of-a-kind a guaranteed score of 15 but keeping 66 maximizes the expected average score The situations where the strategy to maximize the average score differs are all after the first throw and are as follows: For example, with the player should keep 6 rather than 22 and with they should keep not If they follow the strategy to maximize the average score they will get a three-of-a-kind As with three-of-a-kind there are slightly different strategies depending on whether a player is simply trying to get a four-of-a-kind or he is trying to maximize his average score.

Different strategies will also be required should he need to achieve a specific target. The strategy to maximize his chance of getting a four-of-a-kind involves keeping any four-of-a-kind that he has.

If he has a four-of-a-kind then after the first throw he will keep the other if it is a 5 and 6, while after the second throw he will keep it if it is a 4, 5 or 6.

So that with he keeps and will throw the 3. If he does not have a four-of-a-kind, the player should keep any three-of-a-kind or pair that he has and re-roll the other dice.

With two pairs he will keep the higher pair. With no pair he will keep the highest die. Following this strategy gives him a As with three-of-a-kind this strategy does not maximize the average score since there are a few situations after the first throw, where it is better to keep other combinations.

For instance, after throwing , keeping maximizes the chance of getting a four-of-a-kind but keeping 66 maximizes the expected average score 6.

Following the strategy to maximize the average score he will get a four-of-a-kind A player will keep a Yahtzee or Full House. A Yahtzee will score 25 under the Joker rule, even though it is not strictly a full house.

He will keep any four-of-a-kind, three-of-a-kind or pairs that are thrown and re-roll the others. With two pairs he will keep both.

On average he will succeed The strategy is complicated by the fact that, because of the Joker rule, the player will score 30 if he gets a Yahtzee.

Clearly he keeps any Small Straight or Yahtzee that he throws. After the first throw he will keep a run of 3 or 3 out of 4 e.

Otherwise, he will keep a 3 or 4 or both and a 2 or 5 if he also has a 3 or 4 e. He will not keep just 2, 5 or The only difference after the second throw is that he keeps , , and and will try to throw a Yahtzee unless the other die is a 3 or 4.

Again the strategy is complicated by the fact that, because of the Joker rule, he will score 40 if he gets a Yahtzee.

Clearly he will keep any Large Straight or Yahtzee that he throws. The best strategy is also to keep a four-of-a-kind and try to throw a Yahtzee, even after the first throw.

The player should not keep a three-of-a-kind. Keep a small straight or 4 out of 5 e. Otherwise he should simply keep any 2, 3, 4 or 5 just one of each , so that with he would keep A player should keep any Yahtzee, four-of-a-kind, three-of-a-kind or pair that is thrown and re-roll the others.

On average he will succeed 4. To get the maximum average score the strategy is straightforward. After the first throw the player will keep any 5s and 6s.

After the second throw he will keep any 4s, 5s, and 6s. On average he will score Different strategies will be required when he needs to achieve a specific target.

The strategy for maximizing the expected score has been determined. It is important to note that the "Optimal" strategy simply maximises the average score.

It does not maximise the chances of winning a game. There are two main reasons for this. Firstly, the Optimal strategy takes no account of any opponents.

In normal gameplay a player will adjust their strategy depending on the scores of the other player or players. Secondly, the Optimal strategy tends to give undue importance to Yahtzee bonuses.

It is rare for a player without a Yahtzee bonus to beat a player with one. So, in normal gameplay, a players' strategy is not significantly influenced by the value given for a Yahtzee bonus.

Consider the situation where the Yahtzee bonus was worth a million rather than a hundred. It would not influence normal gameplay where the objective is to score more than the opponent.

It would, however, affect the "Optimal" strategy since scoring a million would have a dramatic effect on the average score. The "Optimal" strategy would be dominated by the prospect of a Yahtzee bonus.

Even with a Yahtzee bonus worth the "Optimal" strategy tends to give too much importance to Yahtzee bonuses.

Despite these limitations the "Optimal" strategy does provide a useful guide as to the best strategy, especially in the early rounds.

The "Optimal" strategy for the first round is described in the next section. After the first round, the number of different games rapidly makes detailed analysis difficult but, in the early rounds, players generally simply adapt the first round strategy based on the boxes used.

For instance if a player throws in the first round they will keep 66 but if the 6s box was used in the first round and they throw in the second round they will naturally keep 55 instead.

The following table shows the average score obtained using the Optimal strategy and the proportion of the time that zero is scored in a particular category: It is possible to calculate the maximum average score with different rules.

When the rules are changed so that there is no Upper Section bonus the average score drops from This compares with an average bonus score of The average number of Yahtzees thrown in a game is 0.

In about a quarter of these cases the Yahtzee box has a score 0 and so there is no Yahtzee bonus. Although the average score is The 1st Percentile is i.

The strategy for the first round described here is based on that used by the "Optimal" strategy. Sometimes there are two possible plays which are almost equally good.

In these cases the simplest option is given, i. Although there are different ways of throwing 5 dice, because the order of the dice is not important there are only different combinations.

To use the strategy described therefore requires knowing different situations. Three of a Kind 3 dice of the same kind sum of all the dice Four of a Kind 4 dice of the same kind sum of all the dice FullHouse 2 dice of one kind, 3 of another 25 points Small Straight 4 consecutive numbers 30 points Large Straight 5 consecutive numbers 40 points Kniffel 5 dice of the same kind 50 points Chance any combination sum of all the dice In the top section first six rows you get a bonus 35 points in any of the four columns if your subtotal for that column is greater than or equal to When selecting scores you have to proceed scoring in the first column from top to bottom.

Second column is filled from bottom to top and forth can be filled in any order. Third column can be filled in any order but it will score only after the first throw.

If you can't select any score which will give you points you have to mark zero to that scoring row of the first or second column which is next in turn or to any row of the third or fourth column.

Scores List of ten best scoring games is saved on the game server. You can browse this list selecting Scores. When your result is to be added to the list computer asks your name.

Your may enter a nick name or whatever you want to use. You can select an individual game from the window to see the exact points in each combination.

Network Scores Best results are saved on the game server. You can browse the list selecting Network Scores You can select an individual game from the window to see the exact results in each combination If a result from your computers Score list is better than it will be also saved to Network Scores.

Preferences In preferences you can set the user name you want to use when saving your Network Scores. Hints You should normally select combinations in 3rd column when possible.

Wenn man beim Sammeln in der Summe mindestens 63 Punkte beispielsweise für jedes Feld drei Würfel bekommen hat, gibt es einen Bonus von 35 Punkten Jargon: Beim Kniffel sind für zusätzliche Kniffel-Würfe Sonderpunkte vorgesehen.

Wird allerdings ein Yahtzee geworfen, wenn das Yahtzee-Feld bereits mit einer 0 gefüllt ist, entfallen die Yahtzee-Punktzahl und der -Bonus, der Würfelwurf wird wie ein normaler Zahlenwurf mit fünf gleichen Zahlen als Pasch, Full House oder Chance vermerkt.

Ist der Spielzettel voll, so ist das Spiel beendet und die Punkte vom Sammeln und den anderen Spielen und eventuell der Bonus werden zusammengezählt.

Es gewinnt der Spieler mit den meisten Punkten. Der Spielblock ist bei Yahtzee und Kniffel gleich aufgebaut und besteht aus einem oberen und einem unteren Block sowie Zeilen für insgesamt sechs Spiele.

Die folgende Vorlage ist beispielhaft für zwei Spiele ausgefüllt. Das Spiel ist von seinem Spielaufbau sehr einfach und kann auch gut mit Jugendlichen gespielt werden.

Komplexer ist die Entscheidung, welche Felder auszufüllen sind, und die strategische Abschätzung der Wahrscheinlichkeit, bestimmte Würfelfiguren zu bekommen.

Zum Ende des Spiels nimmt die Anzahl der zur Verfügung stehenden Möglichkeiten ab und die Wahlmöglichkeiten werden zunehmend eingeschränkt.

Neben den offiziellen Regeln bietet das Spiel zahlreiche Möglichkeiten, das Spiel durch abgewandelte Regeln abwechslungsreicher zu machen.

Neben diesen leichten Veränderungen des Grundregelwerks gibt es allerdings auch einige Varianten, die den Charakter eines eigenen Spiels haben.

Die Variante Yams arbeitet mit vier Spalten, wobei die erste wie Kniffel funktioniert, die zweite und dritte müssen der Reihe nach gefüllt werden, in der vierten ist eine verpflichtende Ansage nach dem 1.

Kniffel chance -

Wenn ich die Bedingung erfülle, dann darf ich folgende Punktzahl schreiben: Variationen sind ja im Gegensatz zu Kombinationen gleich wahrscheinlich und deshalb für die Berechnung von Wahrscheinlichkeiten geeignet. Sind noch alle Felder frei, so lässt der perfekte Spieler nach dem ersten Wurf liegen. Bei einen Kniffel aus Fünfen oder Sechsen wird alles behalten und man ist am Ziel. Ganz beliebt unter uns wenn z. Dasselbe gilt für die 5. Ausgenommen sind die Fälle , , und

Kniffel Chance Video

Ich kniffel mir Dir. Staybet casino bonus code, die Spiele eines bestimmten Spiel-Anbieters bevorzugen, können von der Startseite über Direktlinks u. If a player throws a Yahtzee but the Yahtzee category has already been used, special rules apply. After the second throw he will kniffel chance any 4s, 5s, and 6s. Don't like the theme? The score of 1, requires using the "Joker" rules for the full house, small straight, and large straight categories. In each turn a player may throw the dice up to three times. There are a large number of versions of Yahtzee fußballspiele heute em can be played online or are available to download. So, before the second roll he will never keep two pairs, he will never keep a pair of 1s and he will never throw all the dice again. The 1st Percentile is i. So, he puts 1,3,4 to the side and texas poker online game throws 3 and sunenergy casino again, hoping to get 2 and 5. They can re-roll a die for the third roll that was not rolled on the second roll. Now he online free deposit casinos keep the Full House if the three-of-a-kind is 2 or 3 and keep two pairs when they are 1 and 2 or 1 and 3. November um Bei zwei Zwillingen und einem Einling wird nur der Zwilling mit der höchsten Augenzahl behalten. Sie gibt an, in wieviel Prozent der Fälle ein Ereignis durchschnittlich eintritt. Sie gibt an, in wieviel Prozent der Fälle ein Ereignis durchschnittlich eintritt. Im englischsprachigen Raum ist zudem die Version Kismet bekannt. Zum leichteren Verständnis der Berechnungen rechne ich ein Beispiel eines vereinfachten Spiels komplett durch. Spielanleitung und Spielverlauf von Kniffel. Daraus ergibt sich eine mittlere Punktzahl von 10, In Klammern steht jeweils die Anzahl der Strategien, die man durch unterschiedliches Behalten verfolgen kann. Bei zwei Einlingen wird eine eventuell vorhandene Eins oder Sechs verworfen. Klicke hier, um einen Kommentar casino tschechien grenze philippsreut schreiben. Die deterministischen Entscheidungen, also welche Würfel ich book of dead casinomeister lasse, werden einfach nach dem höchsten Erwartungswert gefällt. Diese müssen, beispielsweise mit "cc -o kniffel hotel melia coral umag casino. Bei einem Zwilling und drei Einlingen wird nur der Zwilling behalten, wenn es sich dabei um Vieren, Fünfen Beste Spielothek in Teilhof finden Sechsen handelt. Diese müssen, beispielsweise mit "cc -o kniffel kniffel. Auf den Punkt gebracht 1. Ist das Spiel vorbei, ist der Erwartungswert die erreichte Punktzahl. Auf diese Weise kann ich Schritt für Schritt online casion Zustände von rechts nach links berechnen und klaver casino zu einem Erwartungswert vor dem Spiel a priori von 65,31 Punkten. Daher dauert der erste Programmaufruf etwa eine halbe Stunde, bei älteren Rechnern mehrere Stunden. Bei drei Einlingen kann eine eventuell vorhandene Eins oder Sechs entweder behalten oder verworfen werden. Bei der optimalen Strategie wird sowohl nach dem ersten als auch nach dem zweiten Wurf nur ein Mehrling behalten und es gelten die folgenden unmittelbar einleuchtenden Regeln: Oder willst Du erstmal die Webversion ausprobieren? Das 1er-Feld kann frei sein oder mit 0, 1, 2, 3, 4 oder 5 belegt. Die aktuelle Kniffel-Runde gewinnt, wer die meisten Punkte gewürfelt hat. Sind noch alle Felder frei, so lässt der perfekte Spieler nach dem ersten Wurf liegen. Sollten während der Spielrunde weitere Kniffel geworden werden so kann dieser wie eine Art Joker behandelt werden und darf in ein beliebiges Feld eingetragen werden. Dagegen wird man sofort aufhören, wenn man eine 5 oder 6 bekommen hat. Kniffel kann im Prinzip von beliebig vielen Spielern gespielt werden. Um möglichst oft einen Viererpasch zu erzielen, muss man eine andere Strategie verfolgen. Ich wurde gefragt, wie wahrscheinlich es ist, mit drei Würfen einen Kniffel zu würfeln. Die folgende Zusammenstellung enthält die 15 möglichen Fälle zum Erzielen eines Kniffels mit den entsprechenden Einzelwahrscheinlichkeiten p bei optimaler Strategie Die Einzelwahrscheinlichkeiten ergeben sich durch stochastische Überlegungen. Ob der Kniffel gewürfelt wurde oder gestrichen ist unerheblich. Viererpasch Mindestens vier mal die selbe Zahl. Für den Erwartungswert und die optimale Strategie ist es ja egal, ob man 63 oder mehr Bonuspunkte erzielt hat.

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