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Balloon Game

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Introduction

The Balloon Game is a keno-based game found on multi-game machines made by U1 Gaming. I have seen such machines at the Las Vegas Club, Rampart, and Dotty's casinos in Las Vegas. Balloon Game is similar to Caveman Keno, except it uses five balloons instead of three eggs.
 

All games by U1 feature random bonuses in addition to the base game. These random bonuses evidently add 0.5% to the total return. I can tell because in all but one case, my return is 0.5% less than the advertised return of the game. This page endeavors to only analyze the base game, and ignores the random bonuses.

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Rules

  1. Player picks 2 to 10 numbers from 1 to 80.
  2. The game will pick 5 numbers from the remaining numbers and mark them with balloons.
  3. The game will pick 20 numbers from 1 to 80. As the game picks numbers, it will count the number that match the player's picks as well as the balloons. When the game picks a number marked with a balloon, that balloon will pop.
  4. The player is paid according to the number of player numbers caught and balloons popped.

Odds

Following is an analysis of the game according to how many numbers the player chooses, from 2 to 10. Rather than repeat the same table introductions for each number of picks, let me explain them here.

Pay Table: How much each win pays. These wins are based on a maximum $2 bet as found at the Rampart casino. As the player bets less, the pay table will be depressed.
Combinations: Number of combinations for each win, according to the number of catches and balloons popped. Please note that Excel, where I did the math, can handle only 15 significant digits. Should anyone at Microsoft read this, please do something about this.
Probabilities: Probability of each win, according to the number of catches and balloons popped.
Return: Contribution to the return for each win, according to the number of catches and balloons popped. The lower image--rightell shows the overall return of the game.

Pick 2

Pick 2 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
2 7 14 30 120 530
1 0 1 2 10 80
0 0 0 0 3.5 60

Pick 2 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
2 143,371,172,961,555,000 52,717,590,634,746,100 14,542,783,623,378,200 1,848,658,935,175,200 86,270,750,308,176 212,566,476,905,162,000
1 868,289,728,101,701,000 353,517,960,727,121,000 105,435,181,269,492,000 14,542,783,623,378,200 739,463,574,070,080 1,342,525,117,295,760,000
0 1,225,735,666,170,230,000 549,916,827,797,744,000 176,758,980,363,561,000 26,358,795,317,373,100 1,454,278,362,337,820 1,980,224,548,011,250,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 2 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
2 0.0405539892882 0.0149117047851 0.0041135737338 0.0005229119153 0.0000244025560 0.0601265822785
1 0.2456045494020 0.0999961379708 0.0298234095702 0.0041135737338 0.0002091647661 0.3797468354430
0 0.3467117555725 0.1555495479546 0.0499980689854 0.0074558523926 0.0004113573734 0.5601265822785
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 2 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
2 0.283878 0.208764 0.123407 0.062749 0.012933 0.691732
1 0.000000 0.099996 0.059647 0.041136 0.016733 0.217512
0 0.000000 0.000000 0.000000 0.026095 0.024681 0.050777
Total 0.283878 0.308760 0.183054 0.129981 0.054348 0.960021

Pick 3

Pick 3 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
3 20 50 120 305 1,000
2 1 2 10 30 100
1 - - 2 6 80
0 - - - 4 60

Pick 3 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
3 34,141,118,125,016,500 11,554,540,413,095,000 2,988,243,210,283,200 354,537,330,033,600 15,363,284,301,456 49,053,802,362,729,800
2 327,690,164,509,615,000 123,489,150,664,953,000 34,663,621,239,285,100 4,482,364,815,424,800 212,722,398,020,160 490,538,023,627,298,000
1 974,744,427,642,937,000 406,787,790,425,728,000 123,489,150,664,953,000 17,331,810,619,642,600 896,472,963,084,960 1,523,249,652,316,350,000
0 900,820,856,955,922,000 414,320,897,655,834,000 135,595,930,141,909,000 20,581,525,110,825,500 1,155,454,041,309,500 1,472,474,663,905,800,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 3 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
3 0.0096571612698 0.0032683188570 0.0008452548768 0.0001002844769 0.0000043456607 0.0138753651412
2 0.0926904840551 0.0349301577843 0.0098049565710 0.0012678823152 0.0000601706861 0.1387536514119
1 0.2757163400479 0.1150640491719 0.0349301577843 0.0049024782855 0.0002535764630 0.4308666017527
0 0.2548063088899 0.1171948648973 0.0383546830573 0.0058216929641 0.0003268318857 0.4165043816943
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 3 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
3 0.193143 0.163416 0.101431 0.030587 0.004346 0.492922
2 0.092690 0.069860 0.098050 0.038036 0.006017 0.304654
1 0.000000 0.000000 0.069860 0.029415 0.020286 0.119561
0 0.000000 0.000000 0.000000 0.023287 0.019610 0.042897
Total 0.285834 0.233276 0.269340 0.121325 0.050259 0.960034

Pick 4

Pick 4 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
4 45 100 200 500 2,400
3 4 10 20 60 200
2 1 2 4 10 100
1 - - 1 4 60
0 - - - 3 80

Pick 4 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
4 7,775,242,819,645,210 2,407,195,919,394,800 581,047,290,888,400 64,013,684,589,400 2,560,547,383,576 10,830,060,261,901,400
3 105,463,501,221,485,000 36,589,377,974,801,000 9,628,783,677,579,200 1,162,094,581,776,800 51,210,947,671,520 152,894,968,403,314,000
2 497,185,077,187,001,000 192,094,234,367,705,000 54,884,066,962,201,400 7,221,587,758,184,400 348,628,374,533,040 751,733,594,649,625,000
1 968,202,518,732,581,000 414,320,897,655,834,000 128,062,822,911,803,000 18,294,688,987,400,500 962,878,367,757,920 1,529,843,806,655,380,000
0 658,770,227,272,777,000 310,740,673,241,876,000 103,580,224,413,959,000 16,007,852,863,975,400 914,734,449,370,024 1,090,013,712,241,960,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 4 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
4 0.0021993062309 0.0006808997619 0.0001643551149 0.0000181069194 0.0000007242768 0.0030633923039
3 0.0298314201557 0.0103496763805 0.0027235990475 0.0003287102299 0.0000144855356 0.0432478913492
2 0.1406338378768 0.0543358009978 0.0155245145708 0.0020426992856 0.0000986130690 0.2126354658000
1 0.2738658948127 0.1171948648973 0.0362238673319 0.0051748381903 0.0002723599048 0.4327318251369
0 0.1863398351867 0.0878961486730 0.0292987162243 0.0045279834165 0.0002587419095 0.3083214254100
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 4 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
4 0.098969 0.068090 0.032871 0.009053 0.001738 0.210722
3 0.119326 0.103497 0.054472 0.019723 0.002897 0.299914
2 0.140634 0.108672 0.062098 0.020427 0.009861 0.341692
1 0.000000 0.000000 0.036224 0.020699 0.016342 0.073265
0 0.000000 0.000000 0.000000 0.013584 0.020699 0.034283
Total 0.358928 0.280258 0.185665 0.083486 0.051538 0.959876

Pick 5

Pick 5 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
5 200 400 1,000 2,000 5,000
4 15 30 60 200 500
3 2 5 13 35 200
2 - 1 2 5 100
1 - - - 2 60
0 - - - 3 80

Pick 5 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
5 1,687,749,477,006,660 474,658,350,303,200 106,388,940,585,200 10,819,214,296,800 396,704,524,216 2,280,012,686,716,080
4 30,437,466,713,192,700 9,662,687,845,458,000 2,373,291,751,516,000 265,972,351,463,000 10,819,214,296,800 42,750,237,875,926,500
3 202,783,819,627,327,000 72,148,069,246,086,400 19,325,375,690,916,000 2,373,291,751,516,000 106,388,940,585,200 296,736,945,256,431,000
2 625,857,975,684,341,000 248,008,988,033,422,000 72,148,069,246,086,400 9,662,687,845,458,000 474,658,350,303,200 956,152,379,159,611,000
1 897,324,160,573,556,000 393,896,628,053,082,000 124,004,494,016,711,000 18,037,017,311,521,600 966,268,784,545,800 1,434,228,568,739,420,000
0 479,305,395,158,066,000 231,961,347,631,259,000 78,779,325,610,616,400 12,400,449,401,671,100 721,480,692,460,864 803,167,998,494,073,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 5 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
5 0.0004773970443 0.0001342619249 0.0000300931901 0.0000030603244 0.0000001122119 0.0006449246956
4 0.0086095459328 0.0027331891850 0.0006713096244 0.0000752329751 0.0000030603244 0.0120923380417
3 0.0573594585237 0.0204078125813 0.0054663783700 0.0006713096244 0.0000300931901 0.0839350522895
2 0.1770302712709 0.0701518557484 0.0204078125813 0.0027331891850 0.0001342619249 0.2704573907106
1 0.2538172328804 0.1114176532474 0.0350759278742 0.0051019531453 0.0002733189185 0.4056860860658
0 0.1355763886107 0.0656126180235 0.0222835306495 0.0035075927874 0.0002040781258 0.2271842081969
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 5 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
5 0.095479 0.053705 0.030093 0.006121 0.000561 0.185959
4 0.129143 0.081996 0.040279 0.015047 0.001530 0.267994
3 0.114719 0.102039 0.068330 0.023496 0.006019 0.314602
2 0.000000 0.070152 0.040816 0.013666 0.013426 0.138060
1 0.000000 0.000000 0.000000 0.010204 0.016399 0.026603
0 0.000000 0.000000 0.000000 0.010523 0.016326 0.026849
Total 0.339342 0.307891 0.179517 0.079056 0.054261 0.960067

Pick 6

Pick 6 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
6 500 1,200 2,500 6,000 15,000
5 22 50 111 475 1,500
4 5 10 20 50 300
3 2 4 8 20 200
2 - - 2 6 100
1 - - - 3 60
0 - - - 5 80

Pick 6 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
6 347,856,762,436,488 88,150,836,484,880 18,238,104,100,320 1,700,162,246,640 56,672,074,888 456,002,537,343,216
5 8,039,356,287,421,060 2,319,045,082,909,920 528,905,018,909,280 54,714,312,300,960 2,040,194,695,968 10,944,060,896,237,200
4 71,214,009,421,025,500 23,190,450,829,099,200 5,797,612,707,274,800 661,131,273,636,600 27,357,156,150,480 100,890,561,387,187,000
3 310,615,626,693,288,000 113,375,537,386,707,000 30,920,601,105,465,600 3,865,075,138,183,200 176,301,672,969,760 458,953,141,996,613,000
2 705,825,243,506,547,000 286,981,829,010,103,000 85,031,653,040,030,400 11,595,225,414,549,600 579,761,270,727,480 1,090,013,712,241,960,000
1 794,458,895,285,649,000 357,883,222,059,657,000 114,792,731,604,041,000 17,006,330,608,006,100 927,618,033,163,968 1,285,068,797,590,520,000
0 346,895,579,277,124,000 172,314,143,954,650,000 59,647,203,676,609,600 9,566,060,967,003,420 566,877,686,933,536 588,989,865,562,320,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 6 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
6 0.0000983948107 0.0000249343575 0.0000051588326 0.0000004809081 0.0000000160303 0.0001289849391
5 0.0022740134019 0.0006559654044 0.0001496061449 0.0000154764977 0.0000005770897 0.0030956385387
4 0.0201436042935 0.0065596540440 0.0016399135110 0.0001870076811 0.0000077382489 0.0285379177784
3 0.0878607779894 0.0320694197707 0.0087462053920 0.0010932756740 0.0000498687150 0.1298195475411
2 0.1996498234143 0.0811757187946 0.0240520648280 0.0032798270220 0.0001639913511 0.3083214254100
1 0.2247207500907 0.1012308963791 0.0324702875178 0.0048104129656 0.0002623861618 0.3634947331150
0 0.0981229302622 0.0487408019603 0.0168718160632 0.0027058572932 0.0001603470989 0.1666017526777
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 6 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
6 0.049197 0.029921 0.012897 0.002885 0.000240 0.095142
5 0.050028 0.032798 0.016606 0.007351 0.000866 0.107650
4 0.100718 0.065597 0.032798 0.009350 0.002321 0.210785
3 0.175722 0.128278 0.069970 0.021866 0.009974 0.405808
2 0.000000 0.000000 0.048104 0.019679 0.016399 0.084182
1 0.000000 0.000000 0.000000 0.014431 0.015743 0.030174
0 0.000000 0.000000 0.000000 0.013529 0.012828 0.026357
Total 0.375665 0.256594 0.180375 0.089092 0.058371 0.960098

Pick 7

Pick 7 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
7 2,500 5,000 10,000 20,000.0 25,000
6 110 220 500 1,000 4,000
5 10 20 55 120 500
4 2 6 15 50 300
3 1 2 4 10 200
2 - - 2 5 100
1 - - - 3 60
0 - - - 6 80

Pick 7 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
7 67,778,853,872,488 15,330,580,258,240 2,907,523,842,080 246,400,325,600 7,392,009,768 86,270,750,308,176
6 1,960,545,359,948,000 509,741,793,586,480 107,314,061,807,680 10,176,333,447,280 344,960,455,840 2,588,122,509,245,280
5 22,256,110,926,129,700 6,587,432,409,425,280 1,529,225,380,759,440 160,971,092,711,520 6,105,800,068,368 30,539,845,609,094,300
4 129,072,503,772,177,000 43,131,997,918,856,000 10,979,054,015,708,800 1,274,354,483,966,200 53,657,030,903,840 184,511,567,221,611,000
3 414,504,842,941,077,000 155,275,192,507,882,000 43,131,997,918,856,000 5,489,527,007,854,400 254,870,896,793,240 618,656,431,272,462,000
2 739,452,435,144,519,000 308,609,445,109,415,000 93,165,115,504,728,900 12,939,599,375,656,800 658,743,240,942,528 1,154,825,338,375,260,000
1 680,384,566,118,417,000 314,660,610,699,795,000 102,869,815,036,472,000 15,527,519,250,788,200 862,639,958,377,120 1,114,305,151,063,850,000
0 249,697,784,117,350,000 127,362,628,140,393,000 44,951,515,814,256,500 7,347,843,931,176,540 443,643,407,165,376 429,803,415,410,342,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 7 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
7 0.0000191719357 0.0000043364100 0.0000008224226 0.0000000696968 0.0000000020909 0.0000244025560
6 0.0005545601245 0.0001441856324 0.0000303548700 0.0000028784790 0.0000000975756 0.0007320766814
5 0.0062953665332 0.0018633220183 0.0004325568971 0.0000455323050 0.0000017270874 0.0086385048410
4 0.0365094657960 0.0122003227389 0.0031055366972 0.0003604640809 0.0000151774350 0.0521909667479
3 0.1172468956855 0.0439211618599 0.0122003227389 0.0015527683486 0.0000720928162 0.1749932414489
2 0.2091616153688 0.0872933091965 0.0263526971159 0.0036600968217 0.0001863322018 0.3266540507047
1 0.1924536699829 0.0890049427101 0.0290977697322 0.0043921161860 0.0002440064548 0.3151925050659
0 0.0706295488361 0.0360258101446 0.0127149918157 0.0020784121237 0.0001254890339 0.1215742519540
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 7 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
7 0.047930 0.021682 0.008224 0.001394 0.000052 0.079282
6 0.061002 0.031721 0.015177 0.002878 0.000390 0.111169
5 0.062954 0.037266 0.023791 0.005464 0.000864 0.130338
4 0.073019 0.073202 0.046583 0.018023 0.004553 0.215380
3 0.117247 0.087842 0.048801 0.015528 0.014419 0.283837
2 0.000000 0.000000 0.052705 0.018300 0.018633 0.089639
1 0.000000 0.000000 0.000000 0.013176 0.014640 0.027817
0 0.000000 0.000000 0.000000 0.012470 0.010039 0.022510
Total 0.362151 0.251714 0.195282 0.087234 0.063591 0.959972

Pick 8

Pick 8 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
8 6,000 12,000 25,000 25,000 25,000
7 400 1,200 2,400 5,000.0 10,000
6 60 120 240 500 1,000
5 6 15 30 60 400
4 2 4 8 18 200
3 - 1 2 5 100
2 - - 1 3 80
1 - - - 4 60
0 - - - 7 100

Pick 8 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
8 12,422,279,003,368 2,479,946,806,480 427,577,035,600 32,611,807,800 869,648,208 15,363,284,301,456
7 442,852,598,952,960 102,805,067,614,080 19,839,574,451,840 1,710,308,142,400 52,178,892,480 567,259,728,053,760
6 6,292,197,343,456,640 1,679,149,437,696,640 359,817,736,649,280 34,719,255,290,720 1,197,215,699,680 8,367,080,988,792,960
5 46,765,234,449,432,600 14,208,187,549,740,800 3,358,298,875,393,280 359,817,736,649,280 13,887,702,116,288 64,705,426,313,332,200
4 199,688,464,482,562,000 68,503,761,400,536,000 17,760,234,437,176,000 2,098,936,797,120,800 89,954,434,162,320 288,141,351,551,558,000
3 503,456,977,119,673,000 193,637,298,892,182,000 54,803,009,120,428,800 7,104,093,774,870,400 335,829,887,539,328 759,337,208,794,693,000
2 734,208,091,632,856,000 314,660,610,699,795,000 96,818,649,446,090,900 13,700,752,280,107,200 710,409,377,487,040 1,160,098,513,436,340,000
1 567,808,620,811,661,000 269,709,094,885,539,000 89,903,031,628,513,000 13,831,235,635,155,800 782,900,130,291,840 942,034,883,091,160,000
0 178,721,706,515,893,000 93,648,991,279,701,000 33,713,636,860,692,400 5,618,939,476,782,060 345,780,890,878,896 312,049,055,023,947,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 8 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
0 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000
0 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000 0.0000000000000
8 0.0000035137675 0.0000007014781 0.0000001209445 0.0000000092246 0.0000000002460 0.0000043456607
7 0.0001252653458 0.0000290794553 0.0000056118247 0.0000004837780 0.0000000147593 0.0001604551631
6 0.0017798117878 0.0004749644360 0.0001017780934 0.0000098206932 0.0000003386446 0.0023667136551
5 0.0132280205131 0.0040189298434 0.0009499288721 0.0001017780934 0.0000039282773 0.0183025855993
4 0.0564839059507 0.0193769831735 0.0050236623042 0.0005937055450 0.0000254445234 0.0815037014968
3 0.1424079083362 0.0547722724370 0.0155015865388 0.0020094649217 0.0000949928872 0.2147862251209
2 0.2076781996570 0.0890049427101 0.0273861362185 0.0038753966347 0.0002009464922 0.3281456217125
1 0.1606104229356 0.0762899508944 0.0254299836315 0.0039123051741 0.0002214512363 0.2664641138718
0 0.0505532459691 0.0264895662828 0.0095362438618 0.0015893739770 0.0000978076294 0.0882662377200
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 8 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
8 0.021083 0.008418 0.003024 0.000231 0.000006 0.032761
7 0.050106 0.034895 0.013468 0.002419 0.000148 0.101036
6 0.106789 0.056996 0.024427 0.004910 0.000339 0.193460
5 0.079368 0.060284 0.028498 0.006107 0.001571 0.175828
4 0.112968 0.077508 0.040189 0.010687 0.005089 0.246441
3 0.000000 0.054772 0.031003 0.010047 0.009499 0.105322
2 0.000000 0.000000 0.027386 0.011626 0.016076 0.055088
1 0.000000 0.000000 0.000000 0.015649 0.013287 0.028936
0 0.000000 0.000000 0.000000 0.011126 0.009781 0.020906
Total 0.370313 0.292873 0.167995 0.072802 0.055795 0.959779

Pick 9

Pick 9 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
9 8,000 16,000 25,000 25,000 25,000
8 2,000 4,000 8,000 16,000 25,000
7 150 300 600 1,200.0 4,000
6 20 42 100 200 600
5 4 8 19 44 300
4 1 3 7 15 200
3 - - 2 5 100
2 - - 1 3 80
1 - - - 4 60
0 - - - 8 100

Pick 9 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
9 2,128,985,542,008 370,141,314,400 57,435,721,200 3,893,947,200 90,858,768 2,560,547,383,576
8 92,639,641,152,240 18,988,249,428,720 3,331,271,829,600 258,460,745,400 7,009,104,960 115,224,632,260,920
7 1,622,278,130,679,360 386,669,806,548,480 75,952,997,714,880 6,662,543,659,200 206,768,596,320 2,091,770,247,198,240
6 15,091,276,392,118,100 4,135,218,764,476,800 902,229,548,613,120 88,611,830,667,360 3,109,187,040,960 20,220,445,722,916,300
5 82,584,862,923,046,200 25,765,593,840,201,600 6,202,828,146,715,200 676,672,161,459,840 26,583,549,200,208 115,256,540,620,623,000
4 276,854,373,145,566,000 97,541,176,680,763,200 25,765,593,840,201,600 3,101,414,073,357,600 135,334,432,291,968 403,397,892,172,181,000
3 570,615,883,582,465,000 225,428,497,217,764,000 65,027,451,120,508,800 8,588,531,280,067,200 413,521,876,447,680 870,073,885,077,252,000
2 699,432,167,706,901,000 307,951,429,234,981,000 96,612,213,093,327,400 13,934,453,811,537,600 736,159,824,005,760 1,118,666,423,670,750,000
1 463,926,656,486,393,000 226,434,874,437,486,000 76,987,857,308,745,200 12,076,526,636,665,900 696,722,690,576,880 780,122,637,559,867,000
0 127,174,300,239,627,000 68,489,560,786,647,000 25,159,430,493,054,000 4,277,103,183,819,180 268,367,258,592,576 225,368,761,961,739,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 9 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
9 0.0000006022051 0.0000001046982 0.0000000162463 0.0000000011014 0.0000000000257 0.0000007242768
8 0.0000262040614 0.0000053710188 0.0000009422840 0.0000000731082 0.0000000019826 0.0000325924550
7 0.0004588778105 0.0001093734735 0.0000214840752 0.0000018845680 0.0000000584866 0.0005916784138
6 0.0042687204722 0.0011696885365 0.0002552047716 0.0000250647544 0.0000008794651 0.0057195579998
5 0.0233599654461 0.0072880593429 0.0017545328048 0.0001914035787 0.0000075194263 0.0326014805987
4 0.0783110652651 0.0275905103694 0.0072880593429 0.0008772664024 0.0000382807157 0.1141051820955
3 0.1614044856609 0.0637647350759 0.0183936735796 0.0024293531143 0.0001169688537 0.2461092162844
2 0.1978414771329 0.0871071827376 0.0273277436040 0.0039415014813 0.0002082302669 0.3164261352227
1 0.1312263565193 0.0640493990718 0.0217767956844 0.0034159679505 0.0001970750741 0.2206655943001
0 0.0359725396892 0.0193729663859 0.0071165998969 0.0012098219825 0.0000759103989 0.0637478383534
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 9 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
9 0.004818 0.001675 0.000406 0.000028 0.000001 0.006927
8 0.052408 0.021484 0.007538 0.001170 0.000050 0.082650
7 0.068832 0.032812 0.012880 0.002261 0.000234 0.117019
6 0.085374 0.049127 0.025520 0.005013 0.000527 0.165562
5 0.093440 0.058304 0.033336 0.008422 0.002256 0.195758
4 0.078311 0.082772 0.051016 0.013159 0.007656 0.232914
3 0.000000 0.000000 0.036787 0.012147 0.011697 0.060631
2 0.000000 0.000000 0.027328 0.011825 0.016658 0.055811
1 0.000000 0.000000 0.000000 0.013664 0.011825 0.025488
0 0.000000 0.000000 0.000000 0.009679 0.007591 0.017270
Total 0.383183 0.246174 0.194812 0.077366 0.058494 0.960030

Pick 10

Pick 10 Pay Table

Balloons
Catch 0 or 1 2 3 4 5
10 10,000 20,000 25,000 25,000 25,000
9 3,000 6,000 12,000 25,000 25,000
8 300 600 1,200 2,500 5,000
7 60 125 300 599.5 2,000
6 7 15 35 100 500
5 3 6 12 30 300
4 1 2 4 8 200
3 - - 2 6 100
2 - - 1 4 80
1 - - - 5 60
0 - - - 10 100

Pick 10 CombinationsExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
10 338,847,548,728 50,473,815,600 6,961,905,600 412,994,400 8,259,888 396,704,524,216
9 17,901,379,932,800 3,196,674,988,000 504,738,156,000 34,809,528,000 825,988,800 21,638,428,593,600
8 382,641,996,063,600 80,556,209,697,600 14,385,037,446,000 1,135,660,851,000 31,328,575,200 478,750,232,633,400
7 4,387,215,112,761,600 1,074,082,795,968,000 214,816,559,193,600 19,180,049,928,000 605,685,787,200 5,695,900,203,638,400
6 30,050,564,532,962,400 8,458,402,018,247,990 1,879,644,892,944,000 187,964,489,294,400 6,713,017,474,800 40,583,288,950,923,600
5 129,109,048,406,537,000 41,381,105,258,505,600 10,150,082,421,897,600 1,127,786,935,766,400 45,111,477,430,656 181,813,134,500,138,000
4 353,833,081,570,496,000 128,084,373,419,184,000 34,484,254,382,088,000 4,229,201,009,124,000 187,964,489,294,400 520,818,874,870,186,000
3 612,975,215,648,952,000 248,849,639,785,843,000 73,191,070,525,248,000 9,852,644,109,168,000 483,337,258,185,600 945,351,907,327,397,000
2 644,424,503,765,270,000 291,620,671,624,035,000 93,318,614,919,691,200 13,723,325,723,484,000 738,948,308,187,600 1,043,826,064,340,670,000
1 372,268,617,481,488,000 186,789,711,236,310,000 64,804,593,694,230,000 10,368,734,991,076,800 609,925,587,710,400 634,841,582,990,815,000
0 89,947,438,491,477,900 49,810,589,663,016,000 18,678,971,123,631,000 3,240,229,684,711,500 207,374,699,821,536 161,884,603,662,658,000
Total 2,237,396,567,233,490,000 956,152,379,159,611,000 296,736,945,256,431,000 42,750,237,875,926,500 2,280,012,686,716,080 3,535,316,142,212,170,000

Pick 10 ProbabilitiesExpand

Balloons
Catch 0 or 1 2 3 4 5 Total
10 0.0000000958465 0.0000000142770 0.0000000019692 0.0000000001168 0.0000000000023 0.0000001122119
9 0.0000050635867 0.0000009042119 0.0000001427703 0.0000000098462 0.0000000002336 0.0000061206488
8 0.0001082341665 0.0000227861403 0.0000040689536 0.0000003212332 0.0000000088616 0.0001354193553
7 0.0012409682575 0.0003038152043 0.0000607630409 0.0000054252715 0.0000001713244 0.0016111430985
6 0.0085001067300 0.0023925447338 0.0005316766075 0.0000531676608 0.0000018988450 0.0114793945770
5 0.0365198028162 0.0117050650052 0.0028710536805 0.0003190059645 0.0000127602386 0.0514276877050
4 0.1000852730950 0.0362299631113 0.0097542208377 0.0011962723669 0.0000531676608 0.1473188970716
3 0.1733862520327 0.0703896426163 0.0207028360636 0.0027869202393 0.0001367168419 0.2674023677939
2 0.1822820019038 0.0824878624409 0.0263961159811 0.0038817817619 0.0002090190179 0.2952567811057
1 0.1052999512651 0.0528353628707 0.0183306360980 0.0029329017757 0.0001725236339 0.1795713756432
0 0.0254425445627 0.0140894300988 0.0052835362871 0.0009165318049 0.0000586580355 0.0457907007890
Total 0.6328702942627 0.2704573907106 0.0839350522895 0.0120923380417 0.0006449246956 1.0000000000000

Pick 10 Returns

Balloons
Catch 0 or 1 2 3 4 5 Total
10 0.000958 0.000286 0.000049 0.000003 0.000000 0.001296
9 0.015191 0.005425 0.001713 0.000246 0.000006 0.022581
8 0.032470 0.013660 0.004883 0.000803 0.000044 0.051861
7 0.074458 0.037977 0.018229 0.003252 0.000343 0.134259
6 0.059501 0.035888 0.018609 0.005317 0.000949 0.120264
5 0.109559 0.070230 0.034453 0.009570 0.003828 0.227641
4 0.100085 0.072460 0.039017 0.009570 0.010634 0.231766
3 0.000000 0.000000 0.041406 0.016722 0.013672 0.071799
2 0.000000 0.000000 0.026396 0.015527 0.016722 0.058645
1 0.000000 0.000000 0.000000 0.014665 0.010351 0.025016
0 0.000000 0.000000 0.000000 0.009165 0.005866 0.015031
Total 0.392223 0.235926 0.184754 0.084840 0.062414 0.960158

Summary

Here is the return for picking 2 to 10 numbers, again according to a $2 bet made at the Rampart casino.

Summary

Pick Return
2 96.00%
3 96.00%
4 95.99%
5 96.01%
6 96.01%
7 96.00%
8 95.98%
9 96.00%
10 96.02%

For more information about returns for bet amounts other than exactly $2, please visit my page on U1 Gaming theoretical returns.

Written by: Michael Shackleford