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Hallo zusammen,
habe mich wegen der Sonderauslosung bei KENO mal mit dem
Kenospiel befaßt.
Dabei habe ich eine interessante Internetseite gefunden, bei der umfangreiche
Berechnungen dargestellt werden, auch für das deutsche KENO 20 aus 70.
Damit kann man sich eine Menge eigene Berechnungen ersparen:
http://www.johnph77.com/math/kenocomp.html
Hier ein paar Auszüge davon.
Viel Spaß damit, Manfred
(KENO Spiel Berechnungen, 20 Zahlen aus 70 Zahlen gezogen)
Keno Computation Tables - 20 Numbers Drawn From a 70-Number Board
Formula for determining total possibilities - (70!/50!)/(20!) = 161,884,603,662,657,876.
2 Spot (Kenotyp 2 in Deutschland)
Formula for determining total possibilities - (70!/68!)/(2!) = 2,415.
Hits...........Formula.....................................Possibilities.......Odds - 1 :
0............((20!/20!)/(0!))x((50!/48!)/(2!))............1,225.................1.97
1............((20!/19!)/(1!))x((50!/49!)/(1!))............1,000.................2.42
2............((20!/18!)/(2!))x((50!/50!)/(0!))............190...................12.71
(Auszahlungen bei 0 und 1 Richtige gibt es im deutschen KENO nicht)
Im deutschen KENO wird die Wahrscheinlichkeit (Odds)
für 2 Richtige mit 1 : 13 angegeben, anstatt mit 1:12,71
Nach meiner Meinung ist der Kenotyp 2 das sinnvollste Spiel, weil hier die
Gewinnwahrscheinlichkeit am größten und die Auszahlung (sechsfach, nach
Abzug des Einsatzes jedoch fünffach) am höchsten ist.
Wenn man höher gewinnen möchte, muß man eben dieses Spiel entsprechend
vielfach spielen.
Ist nur mal meine persönliche Meinung nach Betrachtung der Gewinnpläne
auf dem deutschen KENO-Sonderschein (von Lotto Baden-Württemberg).
Den Sonderschein sollte sich jeder KENO - Interessierte holen,
denn da ist alles schön tabellarisch zusammengefaßt.
3 Spot (Kenotyp 3)
Formula for determining total possibilities - (70!/67!)/(3!) = 54,740.
Hits............Formula.................................Possibilities........Odds - 1:
0............((20!/20!)/(0!))x((50!/47!)/(3!))............19,600...........2.79
1............((20!/19!)/(1!))x((50!/48!)/(2!))............24,500..........2.23
2............((20!/18!)/(2!))x((50!/49!)/(1!))............9,500............5.76
3............((20!/17!)/(3!))x((50!/50!)/(0!))............1,140............48.02
4 Spot (Kenotyp 4)
Formula for determining total possibilities - (70!/66!)/(4!) = 916,895.
Hits............Formula..................................Possibilities......Odds - 1:
0............((20!/20!)/(0!))x((50!/46!)/(4!))............230,300............3.98
1............((20!/19!)/(1!))x((50!/47!)/(3!))............392,000............2.34
2............((20!/18!)/(2!))x((50!/48!)/(2!))............232,750............3.94
3............((20!/17!)/(3!))x((50!/49!)/(1!))............57,000............16.09
4............((20!/16!)/(4!))x((50!/50!)/(0!))............4,845............189.25
5 Spot (Kenotyp 5)
Formula for determining total possibilities - (70!/65!)/(5!) = 12,013,014.
Hits............Formula............ ......................Possibilities......Odds - 1:
0............((20!/20!)/(0!))x((50!/45!)/(5!))............2,118,760.........5.71
1............((20!/19!)/(1!))x((50!/46!)/(4!))............4,606,000..........2.63
2............((20!/18!)/(2!))x((50!/47!)/(3!))............3,724,000..........3.25
3............((20!/17!)/(3!))x((50!/48!)/(2!))............1,396,500..........8.67
4............((20!/16!)/(4!))x((50!/49!)/(1!))............242,250............49.96
5............((20!/15!)/(5!))x((50!/50!)/(0!))............15,504............780.64
6 Spot (Kenotyp 6 )
Formula for determining total possibilities - (70!/64!)/(6!) = 131,115,985.
Hits............Formula....................................Possibilities.......Odds - 1:
0............((20!/20!)/(0!))x((50!/44!)/(6!))............15,890,700............8.25
1............((20!/19!)/(1!))x((50!/45!)/(5!))............42,375,200............3.09
2............((20!/18!)/(2!))x((50!/46!)/(4!))............43,757,000............3.00
3............((20!/17!)/(3!))x((50!/47!)/(3!))............22,344,000............5.87
4............((20!/16!)/(4!))x((50!/48!)/(2!))............5,935,125............22.09
5............((20!/15!)/(5!))x((50!/49!)/(1!))............775,200............169.14
6............((20!/14!)/(6!))x((50!/50!)/(0!))............38,760............3,382.77
7 Spot (Kenotyp 7 )
Formula for determining total possibilities - (70!/63!)/(7!) = 1,198,774,720.
Hits............Formula....................................Possibilities........Odds - 1:
0............((20!/20!)/(0!))x((50!/43!)/(7!))............99,884,400............12.00
1............((20!/19!)/(1!))x((50!/44!)/(6!))............317,814,000............3.77
2............((20!/18!)/(2!))x((50!/45!)/(5!))............402,564,400............2.98
3............((20!/17!)/(3!))x((50!/46!)/(4!))............262,542,000............4.57
4............((20!/16!)/(4!))x((50!/47!)/(3!))............94,962,000............12.62
5............((20!/15!)/(5!))x((50!/48!)/(2!))............18,992,400............63.12
6............((20!/14!)/(6!))x((50!/49!)/(1!))............1,938,000............618.56
7............((20!/13!)/(7!))x((50!/50!)/(0!))............77,520............15,464.07
8 Spot (Kenotyp 8 )
Formula for determining total possibilities - (70!/62!)/(8!) = 9,440,350,920.
Hits............Formula....................................Possibilities..........Odds - 1:
0............((20!/20!)/(0!))x((50!/42!)/(8!))............536,878,650.......... ..17.58
1............((20!/19!)/(1!))x((50!/43!)/(7!))............1,997,688,000............4.73
2............((20!/18!)/(2!))x((50!/44!)/(6!))............3,019,233,000............3.13
3............((20!/17!)/(3!))x((50!/45!)/(5!))............2,415,386,400............3.91
4............((20!/16!)/(4!))x((50!/46!)/(4!))............1,115,803,500............8.46
5............((20!/15!)/(5!))x((50!/47!)/(3!))............303,878,400............31.07
6............((20!/14!)/(6!))x((50!/48!)/(2!))............47,481,000.............198.82
7............((20!/13!)/(7!))x((50!/49!)/(1!))............3,876,000...............2,435.59
8............((20!/12!)/(8!))x((50!/50!)/(0!))............125,970.................74,941.26
9 Spot (Kenotyp 9)
Formula for determining total possibilities - (70!/61!)/(9!) = 65,033,528,560.
Hits............Formula..................................Possibilities............Odds - 1:
0............((20!/20!)/(0!))x((50!/41!)/(9!))............2,505,433,700............25.96
1............((20!/19!)/(1!))x((50!/42!)/(8!))............10,737,573,000............6.06
2............((20!/18!)/(2!))x((50!/43!)/(7!))............18,978,036,000............3.43
3............((20!/17!)/(3!))x((50!/44!)/(6!))............18,115,398,000............3.59
4............((20!/16!)/(4!))x((50!/45!)/(5!))............10,265,392,200............6.34
5............((20!/15!)/(5!))x((50!/46!)/(4!))............3,570,571,200............18.21
6............((20!/14!)/(6!))x((50!/47!)/(3!))............759,696,000..............85.60
7............((20!/13!)/(7!))x((50!/48!)/(2!))............94,962,000...............684.84
8............((20!/12!)/(8!))x((50!/49!)/(1!))............6,298,500................10,325.24
9............((20!/11!)/(9!))x((50!/50!)/(0!))............167,960...................387,196.53
10 Spot (Kenotyp 10)
Formula for determining total possibilities - (70!/60!)/(10!) = 396,704,524,216.
Hits............Formula................................................Possibilities......Odds - 1::
0............ ((20!/20!)/(0!))x((50!/40!)/(10!)) ............10,272,278,170..........38.62
1............((20!/19!)/(1!))x((50!/41!)/(9!))............50,108,674,000............7.92
2............((20!/18!)/(2!))x((50!/42!)/(8!))............102,006,943,500..........3.89
3............((20!/17!)/(3!))x((50!/43!)/(7!))............113,868,216,000..........3.48
4............((20!/16!)/(4!))x((50!/44!)/(6!)) ............76,990,441,500............5.15
5............((20!/15!)/(5!))x((50!/45!)/(5!))............32,849,255,040............12.08
6............((20!/14!)/(6!))x((50!/46!)/(4!))............8,926,428,000..............44.44
7............((20!/13!)/(7!))x((50!/47!)/(3!))............1,519,392,000...............261.09
8............((20!/12!)/(8!))x((50!/48!)/(2!))............154,313,250.................2,570.77
9............((20!/11!)/(9!))x((50!/49!)/(1!))............8,398,000.....................47,237.98
10............((20!/10!)/(10!))x((50!/50!)/(0!))............184,756...................2,147,180.74
(Es gibt dann noch Berechnungen von Kenotyp 11 bis 20, was jedoch
in Deutschland nicht gespielt werden kann).
Interessante Häufigkeitsverteilungen, z.B. wenn man mit der Hälfte der Zahlen (35) spielt
(besondere Spielvarianten, die in anderen Ländern angeboten werden, z.B. aufteilen der
Zahlen auf dem Spielschein in senkrechte oder waagrechte Hälften, was der Hälfte der
Zahlen entspricht, entsprechend auch für gerade/ungerade Zahlen):
Top/Bottom, Left/Right and Odd/Even
Formula for determining total possibilities - (70!/50!)/(20!) = 161,884,603,662,657,876.
Some games will give the player the option of splitting the playslip in half,
(either horizontally or vertically, depending on how the playslip is configured)
and pay according to the distribution of drawn numbers in any half.
Some games will accomplish the same by dividing the ticket in half according
to whether the drawn numbers are odd or even.
Hits...............Formula.............................................Possibilities...............Odds - 1:
0 of 35.... ((35!/35!)/(0!))x((35!/15!)/(20!))...............3,247,943,160...............49,842,191.10
1 of 35 ...((35!/34!)/(1!))x((35!/16!)/(19!))...............142,097,513,250............1,139,250.08
2 of 35 ...((35!/33!)/(2!))x((35!/17!)/(18!))...............2,699,852,751,750...........59,960.53
3 of 35 ...((35!/32!)/(3!))x((35!/18!)/(17!))...............29,698,380,269,250..........5,450.96
4 of 35 ...((35!/31!)/(4!))x((35!/19!)/(16!))...............212,577,879,822,000.........761.53
5 of 35 ...((35!/30!)/(5!))x((35!/20!)/(15!))...............1,054,386,283,917,120......153.53
6 of 35 ...((35!/29!)/(6!))x((35!/21!)/(14!))...............3,765,665,299,704,000......42.99
7 of 35 ...((35!/28!)/(7!))x((35!/22!)/(13!))...............9,927,663,062,856,000......16.31
8 of 35 ...((35!/27!)/(8!))x((35!/23!)/(12!))...............19,639,507,363,476,000......8.24
9 of 35 ...((35!/26!)/(9!))x((35!/24!)/(11!))...............29,459,261,045,214,000.......5.50
10 of 35...((35!/25!)/(10!))x((35!/25!)/(10!)).............33,701,394,635,724,816.......4.80
11 of 35...((35!/24!)/(11!))x((35!/26!)/(9!))...............29,459,261,045,214,000......5.50
12 of 35...((35!/23!)/(12!))x((35!/27!)/(8!))...............19,639,507,363,476,000......8.24
13 of 35...((35!/22!)/(13!))x((35!/28!)/(7!))...............9,927,663,062,856,000......16.31
14 of 35...((35!/21!)/(14!))x((35!/29!)/(6!))...............3,765,665,299,704,000 ......42.99
15 of 35...((35!/20!)/(15!))x((35!/30!)/(5!))...............1,054,386,283,917,120......153.53
16 of 35...((35!/19!)/(16!))x((35!/31!)/(4!))...............212,577,879,822,000.........761.53
17 of 35...((35!/18!)/(17!))x((35!/32!)/(3!))...............29,698,380,269,250.........5,450.96
18 of 35...((35!/17!)/(18!))x((35!/33!)/(2!))...............2,699,852,751,750..........59,960.53
19 of 35...((35!/16!)/(19!))x((35!/34!)/(1!))...............142,097,513,250............1,139,250.08
20 of 35...((35!/15!)/(20!))x((35!/35!)/(0!))...............3,247,943,160...............49,842,191.10
Note: The above table is deceptive in a way. The formulas, possibilities and
the odds of hitting 9 or 11 spots on top or bottom, left or right, and odd or even
(or combinations like 8-12, 7-13, 6-14, 5-15, 4-16, 3-17, 2-18, 1-19 and 0-20),
are exactly the same.
Therefore the number of possibilities for each of these opposite numbers can be
doubled and the odds halved. The 10, however, is unique.
Top, Middle or Bottom
Formula for determining total possibilities - (70!/50!)/(20!) = 161,884,603,662,657,876.
Some games will give the bettor the option of selecting which area of
the board into which the majority of the 20 drawn numbers will fall or,
as an alternative, allow the bettor to wager that 10 numbers will fall
on either side of the board.
Hits........................Formula............................ Possibilities........................ Odds - 1:
<10 of 20................(Calculated)....................... 64,091,604,513,466,530....... 2.53
10 of 20...((35!/25!)/(10!))x((35!/25!)/(10!))............ 33,701,394,635,724,816....... 4.80
>10 of 20................(Calculated)........................64,091,604,513,466,530...... 2.53
Edge and Inside/Outside
(Wenn man die Keno Zahlen in sieben Reihen mit je 10 Zahlen darstellt,
oder mit 10 Reihen mit je sieben Zahlen, entsprechen die einrahmenden
Zahlen dieses Zahlenblocks 30 verschiedene Zahlen. Für das deutsche Keno
sind die Wahrscheinlichkeiten für 1 bis 10 Treffer aus 30 Zahlen interessant.
Die Berechnungen gelten selbstverständlich auch für jede andere Auswahl von
30 Zahlen aus 70)
The 70-number Keno board can be displayed with seven rows of ten numbers each
or ten rows of seven numbers each. In either case the outer edge (hence the name)
will be composed of 30 numbers.
Formula for determining total possibilities - (70!/50!)/(20!) = 161,884,603,662,657,876.
Hits........ Formula............................................ Possibilities........................Odds - 1::
0 of 30.... ((30!/30!)/(0!))x((40!/20!)/(20!)).............. 137,846,528,820................ 1,174,382.88
1 of 30 ....((30!/29!)/(1!))x((40!/21!)/(19!))..............3,938,472,252,000................41,103.40
2 of 30 ....((30!/28!)/(2!))x((40!/22!)/(18!)) ..............49,320,413,883,000.............3,282.30
3 of 30.... ((30!/27!)/(3!))x((40!/23!)/(17!)) ..............360,253,457,928,000............449.36
4 of 30.... ((30!/26!)/(4!))x((40!/24!)/(16!)) ..............1,722,461,845,718,250.........93.98
5 of 30.... ((30!/25!)/(5!))x((40!/25!)/(15!)) ..............5,732,353,022,550,336.........28.24
6 of 30.... ((30!/24!)/(6!))x((40!/26!)/(14!)) ..............13,779,694,765,746,000.......11.75
7 of 30.... ((30!/23!)/(7!))x((40!/27!)/(13!)).............. 24,497,235,139,104,000........6.61
8 of 30.... ((30!/22!)/(8!))x((40!/28!)/(12!)) ..............32,699,434,404,429,000........4.95
9 of 30.... ((30!/21!)/(9!))x((40!/29!)/(11!)).............. 33,075,289,972,296,000........4.89
10 of 30...((30!/20!)/(10!))x((40!/30!)/(10!)).............25,467,973,278,667,920.......6.36
11 of 30...((30!/19!)/(11!))x((40!/31!)/(9!))..............14,937,227,729,424,000........10.84
12 of 30...((30!/18!)/(12!))x((40!/32!)/(8!))..............6,651,734,223,259,125...........24.34
13 of 30...((30!/17!)/(13!))x((40!/33!)/(7!))..............2,232,749,949,066,000...........72.50
14 of 30...((30!/16!)/(14!))x((40!/34!)/(6!))..............558,187,487,266,500..............290.02
15 of 30...((30!/15!)/(15!))x((40!/35!)/(5!))..............102,068,569,100,160..............1,586.04
16 of 30...((30!/14!)/(16!))x((40!/36!)/(4!))..............13,290,178,268,250...............12,180.77
17 of 30...((30!/13!)/(17!))x((40!/37!)/(3!))..............1,183,227,318,000................136,816.15
18 of 30...((30!/12!)/(18!))x((40!/38!)/(2!))..............67,464,715,500....................2,399,544.75
19 of 30...((30!/11!)/(19!))x((40!/39!)/(1!))..............2,185,092,000......................74,085,944.05
20 of 30...((30!/10!)/(20!))x((40!/40!)/(0!))..............30,045,015..........................5,388,068,658.40
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, alles Gute und viel Glück!
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