Introduction
Circus Keno is a game found on multigame machines made by U1 Gaming. I have seen such machines at the Las Vegas Club, Rampart and Dotty's casinos in Las Vegas. The thrust of Circus Keno is after the player picks his numbers, the game will randomly pick six circus tents from among the remaining numbers. By hitting the circus tents with the 20 drawn numbers, the player can earn multipliers or a free spin bonus.
Rules


The following are the rules of the base game.
 The player chooses 2 to 10 numbers from a range of 1 to 80.
 The game will randomly choose six numbers from among the numbers the player didn't choose. Two will be marked with green tents, two with purple tents, and two with blue tents.
 The player will win in part according to the number of balls that match his chosen numbers. The pay table will depend on the amount bet. Below is an example for a $2 bet.
Pay Table — $2 bet
Catch Pick 2 3 4 5 6 7 8 9 10 10 10000 9 5000 2000 8 2500 1150 178.5 7 1030 344 100 35.5 6 535 97 40 12 7 5 212.5 40 10 6 4 2 4 49.625 12 5 2 2 1 1 3 15.075 3 2 1 1 0 0 0 2 8 2 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  The player can win multipliers according to the following patterns of tents that match the 20ball draw, as follows.
 One pair of tents of one color = 2x multiplier.
 One pair of tents of one color plus one tent of another color = 3x multiplier.
 One pair of tents of one color plus one tent each of the other two colors = 4x multiplier.
 The player can win a free spin bonus round according to the following patterns of tents that match the 20ball draw, as follows.
 Two pairs of tents = Four free spins.
 Five total tents = Seven free spins.
 Six total tents = 20 free spins.
 Finally, there is a random bonus game that adds 0.5025% to the return. This random bonus is not otherwise addressed in this page, because I consider it outside the scope of the Circus game.
Following are the rules for free spins.
 Free spins shall have the same number of marks and wager as the initiating spin.
 Every free spins shall have two bonuses of the following three types:
 Multiplier.
 Extra balls.
 Extra free spins.
 The game shall represent the two bonuses with two spinners, a left one and a right one.
 The game shall choose from two tables of possible wins for both bonuses.
 The game shall then choose the win randomly for each spinner according the following tables. The two wins for each spinner are drawn independently.
 The following tables show the possible wins for both spinners according to the table chosen.
Examples


The image above and to the left was an initial spin. I bet $0.05 and picked ten numbers. I caught four of them, for a base win of $0.05. In addition, I caught both blue tents, one purple tent, and one green tent for a 4x multiplier. Thus, my final win was $0.05 × 4 = $0.20.
The image above and to the right was a bonus spin. On the initial spin I bet $0.05 and picked ten numbers, which carried over to the bonus spins. My two bonuses were three extra balls and a 6x multiplier. Normally, I would have caught five numbers, but one of the three extra balls hit number 25, for a total of six catches andr a base win of $0.35. Then, the 6x multiplier increased my final win for that spin to $0.35 * 6 = $2.10.
Pick 10
The following table shows the return table for the base game given ten numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 10 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

10  10000  184,756  0.00000011  0.00112212 
9  2000  10,077,600  0.00000612  0.01224130 
8  178.5  222,966,900  0.00013542  0.02417235 
7  35.5  2,652,734,400  0.00161114  0.05719558 
6  7  18,900,732,600  0.01147939  0.08035576 
5  2  84,675,282,048  0.05142769  0.10285538 
4  1  242,559,401,700  0.14731890  0.14731890 
3  0  440,275,888,800  0.26740237  0.00000000 
2  0  486,137,960,550  0.29525678  0.00000000 
1  0  295,662,853,200  0.17957138  0.00000000 
0  0  75,394,027,566  0.04579070  0.00000000 
Total  1,646,492,110,120  1.00000000  0.42526139 
The next table summarizes the return of the pick10 game with a $2 bet.
Pick 10 Summary — $2 bet
Win Source  Return 

Base win  0.425261 
Multipliers  0.081177 
Bonus  0.453668 
Total  0.960106 
Pick 9
The following table shows the return table for the base game given nine numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 9 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

9  5000  167,960  0.00000072  0.00362138 
8  1150  7,558,200  0.00003259  0.03748132 
7  100  137,210,400  0.00059168  0.05916784 
6  12  1,326,367,200  0.00571956  0.06863470 
5  4  7,560,293,040  0.03260148  0.13040592 
4  1  26,461,025,640  0.11410518  0.11410518 
3  0  57,072,800,400  0.24610922  0.00000000 
2  0  73,379,314,800  0.31642614  0.00000000 
1  0  51,172,416,900  0.22066559  0.00000000 
0  0  14,783,142,660  0.06374784  0.00000000 
Total  231,900,297,200  1.00000000  0.41341635 
The next table summarizes the return of the pick9 game with a $2 bet.
Pick 9 Summary — $2 bet
Win Source  Return 

Initial spin  0.413416 
Multipliers  0.075515 
Bonus  0.471319 
Total  0.960250 
Pick 8
The following table shows the return table for the base game given eight numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 8 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

8  2500  125,970  0.00000435  0.01086415 
7  344  4,651,200  0.00016046  0.05519658 
6  40  68,605,200  0.00236671  0.09466855 
5  6  530,546,880  0.01830259  0.10981551 
4  2  2,362,591,575  0.08150370  0.16300740 
3  0  6,226,123,680  0.21478623  0.00000000 
2  0  9,512,133,400  0.32814562  0.00000000 
1  0  7,724,138,400  0.26646411  0.00000000 
0  0  2,558,620,845  0.08826624  0.00000000 
Total  28,987,537,150  1.00000000  0.43355219 
The next table summarizes the return of the pick8 game with a $2 bet.
Pick 8 Summary — $2 bet
Win Source  Return 

Initial spin  0.433552 
Multipliers  0.080345 
Bonus  0.446053 
Total  0.959950 
Pick 7
The following table shows the return table for the base game given seven numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 7 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

7  1030  77,520  0.00002440  0.02513463 
6  97  2,325,600  0.00073208  0.07101144 
5  10  27,442,080  0.00863850  0.08638505 
4  2  165,795,900  0.05219097  0.10438193 
3  1  555,903,900  0.17499324  0.17499324 
2  0  1,037,687,280  0.32665405  0.00000000 
1  0  1,001,277,200  0.31519251  0.00000000 
0  0  386,206,920  0.12157425  0.00000000 
Total  3,176,716,400  1.00000000  0.46190629 
The next table summarizes the return of the pick7 game with a $2 bet.
Pick 7 Summary — $2 bet
Win Source  Return 

Initial spin  0.461906 
Multipliers  0.094927 
Bonus  0.403064 
Total  0.959898 
Pick 6
The following table shows the return table for the base game given six numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 6 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

6  535  38,760  0.00012898  0.06900694 
5  40  930,240  0.00309564  0.12382554 
4  5  8,575,650  0.02853792  0.14268959 
3  1  39,010,800  0.12981955  0.12981955 
2  0  92,650,650  0.30832143  0.00000000 
1  0  109,230,240  0.36349473  0.00000000 
0  0  50,063,860  0.16660175  0.00000000 
Total  300,500,200  1.00000000  0.46534162 
The next table summarizes the return of the pick6 game with a $2 bet.
Pick 6 Summary — $2 bet
Win Source  Return 

Initial spin  0.465342 
Multipliers  0.092483 
Bonus  0.401962 
Total  0.959787 
Pick 5
The following table shows the return table for the base game given five numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 5 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

5  212.5  15,504  0.00064492  0.13704650 
4  12  290,700  0.01209234  0.14510806 
3  2  2,017,800  0.08393505  0.16787010 
2  0  6,501,800  0.27045739  0.00000000 
1  0  9,752,700  0.40568609  0.00000000 
0  0  5,461,512  0.22718421  0.00000000 
Total  24,040,016  1.00000000  0.45002466 
The next table summarizes the return of the pick5 game with a $2 bet.
Pick 5 Summary — $2 bet
Win Source  Return 

Initial spin  0.450025 
Multipliers  0.091117 
Bonus  0.418854 
Total  0.959995 
Pick 4
The following table shows the return table for the base game given four numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 4 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

4  49.625  4,845  0.00306339  0.15202084 
3  3  68,400  0.04324789  0.12974367 
2  1  336,300  0.21263547  0.21263547 
1  0  684,400  0.43273183  0.00000000 
0  0  487,635  0.30832143  0.00000000 
Total  1,581,580  1.00000000  0.49439998 
The next table summarizes the return of the pick4 game with a $2 bet.
Pick 4 Summary — $2 bet
Win Source  Return 

Initial spin  0.494400 
Multipliers  0.112217 
Bonus  0.353243 
Total  0.959860 
Pick 3
The following table shows the return table for the base game given three numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 3 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

3  15.075  1,140  0.01387537  0.20917113 
2  2  11,400  0.13875365  0.27750730 
1  0  35,400  0.43086660  0.00000000 
0  0  34,220  0.41650438  0.00000000 
Total  82,160  1.00000000  0.48667843 
The next table summarizes the return of the pick3 game with a $2 bet.
Pick 3 Summary — $2 bet
Win Source  Return 

Initial spin  0.486678 
Multipliers  0.113523 
Bonus  0.359689 
Total  0.959891 
Pick 2
The following table shows the return table for the base game given two numbers picked and a $2 bet, before considering multipliers and the bonus.
Pick 2 Return Table — $2 bet
Catch  Pays  Combinations  Probability  Return 

2  8  190  0.06012658  0.48101266 
1  0  1,200  0.37974684  0.00000000 
0  0  1,770  0.56012658  0.00000000 
Total  3,160  1.00000000  0.48101266 
The next table summarizes the return of the pick2 game with a $2 bet.
Pick 2 Summary — $2 bet
Win Source  Return 

Initial spin  0.481013 
Multipliers  0.115039 
Bonus  0.361551 
Total  0.957603 
Bonus
The following table shows the probability of each tent pattern. It should be noted that actual probabilities will be correlated to the number of catches. These are just overall averages.
Tent Pattern Probabilities
Pattern  Win  Combinations  Probability 

None  Nothing  588,989,865,562,321,000  0.166602 
One  Nothing  1,285,068,797,590,520,000  0.363495 
Two — different colors  Nothing  872,010,969,793,565,000  0.246657 
Three — different colors  Nothing  183,581,256,798,645,000  0.051928 
One pair  2x multiplier  218,002,742,448,391,000  0.061664 
One pair + one singleton  3x multiplier  275,371,885,197,968,000  0.077892 
One pair + two singletons  4x multiplier  80,712,449,109,749,200  0.022830 
Two pair  4 free games  20,178,112,277,437,300  0.005708 
Five total  7 free games  10,944,060,896,237,200  0.003096 
Six total  20 free games  456,002,537,343,216  0.000129 
Total  3,535,316,142,212,170,000  1.000000 
The following table shows the average bonus win per spin according to both table 1 and table 2 (refer to the rules above), as well as the average. The final three columns show the average win per bonus, according to whether it starts with 4, 7 or 20 free spins. The average win per bonus is the product of the average win per spin, initial spins, and 5/3. The reason for the 5/3 is that on average the player will earn 0.666667 extra spins for every initial spin, due to earning more free spins within the bonus.
Average Win per Bonus Spin and Round
Pick  Table 1  Table 2  Average  4 F.G.  7 F.G.  20 F.G. 

2  4.768569  4.446939  4.607754  30.718359  53.757129  153.591797 
3  4.744718  4.423332  4.584025  30.560166  53.480291  152.800832 
4  4.635971  4.367764  4.501867  30.012449  52.521786  150.062246 
5  5.462527  5.213562  5.338045  35.586964  62.277186  177.934818 
6  5.216845  5.028689  5.122767  34.151779  59.765614  170.758897 
7  5.209431  5.064194  5.136813  34.245417  59.92948  171.227085 
8  5.761232  5.608121  5.684676  37.897843  66.321225  189.489214 
9  6.133279  5.880078  6.006679  40.044524  70.077918  200.222622 
10  5.949647  5.613796  5.781721  38.544809  67.453416  192.724045 
Summary
The final table shows the total return according to the number of picks for a $2 bet.
Average Win per Bonus Spin and Round
Pick  Return 

2  95.76% 
3  95.99% 
4  95.99% 
5  96.00% 
6  95.98% 
7  95.99% 
8  95.99% 
9  96.03% 
10  96.01% 
These returns do not include the random bonus, which adds another 0.5025%.
The final pictures above show a U1 Gaming machine on the left and the pay table and return percentages for a $0.05 bet on the right.
For more information about returns for bet amounts other than exactly $2, please visit my page on U1 Gaming theoretical returns.