Bonus Spot Keno

Introduction

Bonus Spot Keno is a keno variant I noticed at the Rampart casino in Las Vegas on January 15, 2016. It plays the same way as conventional keno, except the player is paid more if the first number he chooses matches a ball draw.

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Rules

The player chooses 2 to 10 numbers from the range of 1 to 80.
The first number chosen shall be marked as the "Bonus Spot," but the player may change one of his other choices to the Bonus Spot instead by touching the screen.
The game shall draw 20 numbers at random from 1 to 80.
The player shall be paid, in part, according to how many of his chosen numbers match the 20-ball draw by the game.
If the Bonus Spot matches one of the 20 numbers drawn by the game, then the player will be paid from the Bonus Spot pay table. Otherwise, he will be paid from the Standard pay table, which pays equal or less for every win.

Analysis

The following pay tables are from the 5-cent game at the Rampart casino. The first is the Standard pay table and the second is the Bonus pay table.

Standard Pay Table

Catch	Pick 2	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0
2	15	1	1	0	0	0	0	0	0
3		47	3	2	2	0	0	0	0
4			100	10	2	1	1	0	0
5				820	50	18	10	5	3
6					1,500	240	70	25	14
7						7,500	1,000	250	90
8							10,000	4,000	800
9								10,000	4,000
10									10,000

Bonus Pay Table

Catch	Pick 2	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	0	0	0	0	0	0	0	0
1	0	0	0	0	0	0	0	0	0
2	15	2	2	0	0	0	0	0	0
3		47	7	3	3	1	0	0	0
4			100	12	6	3	2	2	0
5				820	80	30	14	7	8
6					1,500	415	90	47	24
7						7,500	1,900	350	152
8							10,000	5,000	1,000
9								10,000	5,000
10									10,000

The next table shows the weighted average win for each number of picks and catches after considering the probability of matching the Bonus Spot.

Weighted Average Pay Table

Catch	Pick 2	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
1	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00
2	15.00	1.67	1.50	0.00	0.00	0.00	0.00	0.00	0.00
3		47.00	6.00	2.60	2.50	0.43	0.00	0.00	0.00
4			100.00	11.60	4.67	2.14	1.50	0.89	0.00
5				820.00	75.00	26.57	12.50	6.11	5.50
6					1500.00	390.00	85.00	39.67	20.00
7						7500.00	1787.50	327.78	133.40
8							10000.00	4888.89	960.00
9								10000.00	4900.00
10									10000.00

The next table shows the probability of each combination of picks and catches.

Probability

Catch	Pick 2	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0.560127	0.416504	0.308321	0.227184	0.166602	0.121574	0.08826624	0.06374784	0.04579070
1	0.379747	0.430867	0.432732	0.405686	0.363495	0.315193	0.26646411	0.22066559	0.17957138
2	0.060127	0.138754	0.212635	0.270457	0.308321	0.326654	0.32814562	0.31642614	0.29525678
3		0.013875	0.043248	0.083935	0.129820	0.174993	0.21478623	0.24610922	0.26740237
4			0.003063	0.012092	0.028538	0.052191	0.08150370	0.11410518	0.14731890
5				0.000645	0.003096	0.008639	0.01830259	0.03260148	0.05142769
6					0.000129	0.000732	0.00236671	0.00571956	0.01147939
7						0.000024	0.00016046	0.00059168	0.00161114
8							0.00000435	0.00003259	0.00013542
9								0.00000072	0.00000612
10									0.00000011
Total	1.000000	1.000000	1.000000	1.000000	1.000000	1.000000	1.00000000	1.00000000	1.00000000

The final table shows the contribution to the return of each combination of picks and catches, which is the product of the probability and weighted average win tables above. The bottom row shows the overall return.

Return

Catch	Pick 2	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
1	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
2	0.901899	0.231256	0.318953	0.000000	0.000000	0.000000	0.000000	0.000000	0.000000
3		0.652142	0.259487	0.218231	0.324549	0.074997	0.000000	0.000000	0.000000
4			0.306339	0.140271	0.133177	0.111838	0.122256	0.101427	0.000000
5				0.528838	0.232173	0.229537	0.228782	0.199231	0.282852
6					0.193477	0.285510	0.201171	0.226876	0.229588
7						0.183019	0.286814	0.193939	0.214926
8							0.043457	0.159341	0.130003
9								0.007243	0.029991
10									0.001122
Total	0.901899	0.883398	0.884780	0.887341	0.883376	0.884901	0.882479	0.888057	0.888483