# 369 Way Keno

## Introduction

369 Way Keno is a way of playing spot keno with seven different cards at the same time. The player picks three pick-3 cards, all with different picks. The game automatically combines all three ways to combine two of the three cards and plays them as three pick-6 cards, plus one pick-9 game consisting of the numbers picked on the original three cards.

## Rules

1. The player picks nine numbers from 1 to 80.
2. The first three numbers chosen shall form a pick-3 keno card labeled "A."
3. The second three numbers chosen shall form a pick-3 keno card labeled "B."
4. The final three numbers chosen shall form a pick-3 keno card labeled "C."
5. The combined numbers in cards A and B shall form a pick-6 keno card "A+B."
6. The combined numbers in cards A and B shall form a pick-6 keno card "A+B."
7. The combined numbers in cards B and C shall form a pick-6 keno card "B+C."
8. The combined numbers in all three cards shall form a pick-9 keno card "A+B+C."
9. The game shall draw 20 numbers from 1 to 80, without replacement.
10. The game shall score and pay the player according to the seven cards played and the posted pay table.

## Pay Table 1 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 1.

### Pay Table 1

Catch Pick 3 Pick 6 Pick 9
0 1 1 1
1 0 0 0
2 2 0 0
3 11 2 0
4   5 1
5   49 5
6   1,000 40
7     400
8     1,000
9     5,000

The following table shows all possible outcomes of 369 Way keno under pay table 1. The first three columns show the number of catches for cards A, B, and C, respectively. The probability column shows the probability of that combination of catches. The return column is the product of the probability, win, and 1/7. The reason for dividing by 7 is to reflect the return compared to the seven units bet among the seven cards. The lower right cell shows an overall return of 0.847322. ​

### Pay Table 1 Return

Card A Card B Card C Pays Probability Return
0 0 0 7 0.063748 0.063748
0 0 1 3 0.073555 0.031524
0 0 2 5 0.026369 0.018835
0 0 3 18 0.002930 0.007534
0 1 0 3 0.073555 0.031524
0 1 1 1 0.079107 0.011301
0 1 2 5 0.026369 0.018835
0 1 3 20 0.002717 0.007762
0 2 0 5 0.026369 0.018835
0 2 1 5 0.026369 0.018835
0 2 2 11 0.008150 0.012808
0 2 3 70 0.000776 0.007762
0 3 0 18 0.002930 0.007534
0 3 1 20 0.002717 0.007762
0 3 2 70 0.000776 0.007762
0 3 3 1067 0.000068 0.010379
1 0 0 3 0.073555 0.031524
1 0 1 1 0.079107 0.011301
1 0 2 5 0.026369 0.018835
1 0 3 20 0.002717 0.007762
1 1 0 1 0.079107 0.011301
1 1 1 0 0.079107 0.000000
1 1 2 7 0.024451 0.024451
1 1 3 26 0.002329 0.008649
1 2 0 5 0.026369 0.018835
1 2 1 7 0.024451 0.024451
1 2 2 18 0.006986 0.017964
1 2 3 109 0.000613 0.009542
1 3 0 20 0.002717 0.007762
1 3 1 26 0.002329 0.008649
1 3 2 109 0.000613 0.009542
1 3 3 1432 0.000049 0.010087
2 0 0 5 0.026369 0.018835
2 0 1 5 0.026369 0.018835
2 0 2 11 0.008150 0.012808
2 0 3 70 0.000776 0.007762
2 1 0 5 0.026369 0.018835
2 1 1 7 0.024451 0.024451
2 1 2 18 0.006986 0.017964
2 1 3 109 0.000613 0.009542
2 2 0 11 0.008150 0.012808
2 2 1 18 0.006986 0.017964
2 2 2 61 0.001838 0.016021
2 2 3 518 0.000148 0.010946
2 3 0 70 0.000776 0.007762
2 3 1 109 0.000613 0.009542
2 3 2 518 0.000148 0.010946
2 3 3 2122 0.000011 0.003293
3 0 0 18 0.002930 0.007534
3 0 1 20 0.002717 0.007762
3 0 2 70 0.000776 0.007762
3 0 3 1067 0.000068 0.010379
3 1 0 20 0.002717 0.007762
3 1 1 26 0.002329 0.008649
3 1 2 109 0.000613 0.009542
3 1 3 1432 0.000049 0.010087
3 2 0 70 0.000776 0.007762
3 2 1 109 0.000613 0.009542
3 2 2 518 0.000148 0.010946
3 2 3 2122 0.000011 0.003293
3 3 0 1067 0.000068 0.010379
3 3 1 1432 0.000049 0.010087
3 3 2 2122 0.000011 0.003293
3 3 3 8033 0.000001 0.000831
Total       1.000000 0.847322

The next table summarizes the return table above by combining all the different ways to win the same amount.

### Pay Table 1 Return Summary

Win Probability Return
8033 0.000001 0.000831
2122 0.000033 0.009880
1432 0.000148 0.030260
1067 0.000204 0.031137
518 0.000444 0.032838
109 0.003677 0.057254
70 0.004657 0.046574
61 0.001838 0.016021
26 0.006986 0.025948
20 0.016301 0.046574
18 0.029748 0.076494
11 0.024451 0.038423
7 0.137101 0.137101
5 0.237320 0.169514
3 0.220666 0.094571
1 0.237320 0.033903
0 0.079107 0.000000
Total 1.000000 0.847322

The total variance of all seven cards at one unit bet each is 925.34. The standard deviation over all seven units bet is 30.42. The standard deviation based on the total amount bet as one unit is 4.35.

## Pay Table 2 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 2.

### Pay Table 2

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 40 2 0
4   4 1
5   92 3
6   1,500 47
7     352
8     4,700
9     9,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 2 Return Summary

Win Probability Return
0 0 0 0 0.063748 0.000000
0 0 1 0 0.073555 0.000000
0 0 2 2 0.026369 0.007534
0 0 3 44 0.002930 0.018416
0 1 0 0 0.073555 0.000000
0 1 1 0 0.079107 0.000000
0 1 2 4 0.026369 0.015068
0 1 3 47 0.002717 0.018241
0 2 0 2 0.026369 0.007534
0 2 1 4 0.026369 0.015068
0 2 2 9 0.008150 0.010479
0 2 3 139 0.000776 0.015414
0 3 0 44 0.002930 0.018416
0 3 1 47 0.002717 0.018241
0 3 2 139 0.000776 0.015414
0 3 3 1631 0.000068 0.015865
1 0 0 0 0.073555 0.000000
1 0 1 0 0.079107 0.000000
1 0 2 4 0.026369 0.015068
1 0 3 47 0.002717 0.018241
1 1 0 0 0.079107 0.000000
1 1 1 0 0.079107 0.000000
1 1 2 7 0.024451 0.024451
1 1 3 51 0.002329 0.016966
1 2 0 4 0.026369 0.015068
1 2 1 7 0.024451 0.024451
1 2 2 15 0.006986 0.014970
1 2 3 187 0.000613 0.016371
1 3 0 47 0.002717 0.018241
1 3 1 51 0.002329 0.016966
1 3 2 187 0.000613 0.016371
1 3 3 1940 0.000049 0.013665
2 0 0 2 0.026369 0.007534
2 0 1 4 0.026369 0.015068
2 0 2 9 0.008150 0.010479
2 0 3 139 0.000776 0.015414
2 1 0 4 0.026369 0.015068
2 1 1 7 0.024451 0.024451
2 1 2 15 0.006986 0.014970
2 1 3 187 0.000613 0.016371
2 2 0 9 0.008150 0.010479
2 2 1 15 0.006986 0.014970
2 2 2 65 0.001838 0.017071
2 2 3 584 0.000148 0.012341
2 3 0 139 0.000776 0.015414
2 3 1 187 0.000613 0.016371
2 3 2 584 0.000148 0.012341
2 3 3 6466 0.000011 0.010035
3 0 0 44 0.002930 0.018416
3 0 1 47 0.002717 0.018241
3 0 2 139 0.000776 0.015414
3 0 3 1631 0.000068 0.015865
3 1 0 47 0.002717 0.018241
3 1 1 51 0.002329 0.016966
3 1 2 187 0.000613 0.016371
3 1 3 1940 0.000049 0.013665
3 2 0 139 0.000776 0.015414
3 2 1 187 0.000613 0.016371
3 2 2 584 0.000148 0.012341
3 2 3 6466 0.000011 0.010035
3 3 0 1631 0.000068 0.015865
3 3 1 1940 0.000049 0.013665
3 3 2 6466 0.000011 0.010035
3 3 3 13620 0.000001 0.001409
Total       1.000000 0.843210

The total variance of all seven cards at one unit bet each is 3024.29. The standard deviation over all seven units bet is 54.99. The standard deviation based on the total amount bet as one unit is 7.86.

## Pay Table 3 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 3.

### Pay Table 3

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 43 3 0
4   4 1
5   55 6
6   1,600 40
7     300
8     4,700
9     9,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 3 Return Summary

Win Probability Return
13929 0.000001 0.001441
6498 0.000033 0.030255
1994 0.000148 0.042136
1732 0.000204 0.050542
461 0.000444 0.029225
147 0.003677 0.077214
109 0.004657 0.072522
58 0.001838 0.015233
57 0.006986 0.056886
51 0.016301 0.118763
49 0.008790 0.061527
20 0.020958 0.059880
9 0.097804 0.125749
5 0.158213 0.113009
2 0.079107 0.022602
0 0.600840 0.000000
Total 1.000000 0.876984

The total variance of all seven cards at one unit bet each is 3021.98. The standard deviation over all seven units bet is 54.97. The standard deviation based on the total amount bet as one unit is 20.78.

## Pay Table 4 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 4.

### Pay Table 4

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 45 3 0
4   4 1
5   68 6
6   1,500 44
7     300
8     4,700
9     10,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 4 Return Summary

Win Probability Return
14635 0.000001 0.001514
6428 0.000033 0.029929
1898 0.000148 0.040107
1640 0.000204 0.047858
489 0.000444 0.031000
166 0.003677 0.087194
124 0.004657 0.082502
62 0.001838 0.016283
59 0.006986 0.058882
53 0.016301 0.123420
51 0.008790 0.064039
20 0.020958 0.059880
9 0.097804 0.125749
5 0.158213 0.113009
2 0.079107 0.022602
0 0.600840 0.000000
Total 1.000000 0.903968

The total variance of all seven cards at one unit bet each is 2943.71. The standard deviation over all seven units bet is 54.26. The standard deviation based on the total amount bet as one unit is 20.51.

## Pay Table 5 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 5.

### Pay Table 5

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 46 3 0
4   4 1
5   70 6
6   1,600 44
7     335
8     4,700
9     10,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 5 Return Summary

Win Probability Return
14938 0.000001 0.001546
6534 0.000033 0.030423
2035 0.000148 0.043002
1742 0.000204 0.050834
529 0.000444 0.033535
169 0.003677 0.088770
127 0.004657 0.084498
62 0.001838 0.016283
60 0.006986 0.059880
54 0.016301 0.125749
52 0.008790 0.065294
20 0.020958 0.059880
9 0.097804 0.125749
5 0.158213 0.113009
2 0.079107 0.022602
0 0.600840 0.000000
Total 1.000000 0.921054

The total variance of all seven cards at one unit bet each is 3172.37. The standard deviation over all seven units bet is 56.32. The standard deviation based on the total amount bet as one unit is 21.29.

## Pay Table 6 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 6.

### Pay Table 6

Catch Pick 2 Pick 4 Pick 6
0 0 1 1
1 0 0 0
2 3 0 0
3 37 1 0
4   14 1
5   62 8
6   300 65
7     178
8     325
9     375

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 6 Return Summary

Win Probability Return
1386 0.000001 0.000143
826 0.000033 0.003846
580 0.000148 0.012256
441 0.000204 0.012869
359 0.000444 0.022758
182 0.003677 0.095598
116 0.001838 0.030465
111 0.004657 0.073852
73 0.006986 0.072854
53 0.016301 0.123420
40 0.008790 0.050226
30 0.020958 0.089820
21 0.024451 0.073353
6 0.073353 0.062874
4 0.301067 0.172039
1 0.220666 0.031524
0 0.316426 0.000000
Total 1.000000 0.927900

The total variance of all seven cards at one unit bet each is 316.55. The standard deviation over all seven units bet is 17.79. The standard deviation based on the total amount bet as one unit is 6.72.

## Pay Table 7 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 7.

### Pay Table 7

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 48 3 0
4   4 1
5   75 6
6   1,660 44
7     362
8     4,700
9     10,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 7 Return Summary

Win Probability Return
15124 0.000001 0.001565
6608 0.000033 0.030767
2126 0.000148 0.044925
1806 0.000204 0.052702
568 0.000444 0.036008
176 0.003677 0.092447
134 0.004657 0.089155
62 0.008824 0.078160
56 0.016301 0.130406
54 0.008790 0.067806
20 0.020958 0.059880
9 0.097804 0.125749
5 0.158213 0.113009
2 0.079107 0.022602
0 0.600840 0.000000
Total 1.000000 0.945180

The total variance of all seven cards at one unit bet each is 3351.83. The standard deviation over all seven units bet is 57.90. The standard deviation based on the total amount bet as one unit is 21.88.

## Pay Table 8 Analysis

The following table shows what each number of catches pays for pick 3, 6, and 9 keno under what I call pay table 8.

### Pay Table 8

Catch Pick 3 Pick 6 Pick 9
0 0 0 0
1 0 0 0
2 2 0 0
3 48 3 0
4   4 1
5   75 6
6   1,600 44
7     340
8     4,700
9     10,000

The next table shows the probability and contribution to the return for all possible total wins. The return shows the return relative to a seven-unit bet.

### Pay Table 8 Return Summary

Win Probability Return
14944 0.000001 0.001546
6548 0.000033 0.030488
2044 0.000148 0.043193
1746 0.000204 0.050951
546 0.000444 0.034613
176 0.003677 0.092447
134 0.004657 0.089155
62 0.008824 0.078160
56 0.016301 0.130406
54 0.008790 0.067806
20 0.020958 0.059880
9 0.097804 0.125749
5 0.158213 0.113009
2 0.079107 0.022602
0 0.600840 0.000000
Total 1.000000 0.940003

The total variance of all seven cards at one unit bet each is 3217.68. The standard deviation over all seven units bet is 56.72. The standard deviation based on the total amount bet as one unit is 21.44.