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40-Ball Keno


Introduction



Internet casinos using Gamesys software offer a game called "40-Number Instant Keno." It is like conventional keno, except it uses only 40 numbers and the game draws ten balls instead of twenty.

Rules



The player picks 3 to 10 numbers from 1 to 40 and then the game shall pick 10. The more numbers drawn by the casino that match the player's picks, the more he wins.

The following table shows what each number of catches pays by the number of picks.

Pay TableExpand

Catch 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
1 0 0 0 0 0 0 0 0
2 5 2 1 1 0 0 0 0
3 24 10 5 2 2 1 1 0
4 62 25 10 8 4 2 2
5 125 50 25 8 12 8
6 1000 250 250 25 25
7 1000 1000 500 250
8 5000 2500 1250
9 10000 10000
10 20000


Analysis



The following table shows the number of combinations for each number of picks and catches.

Combination TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 4060 27405 142506 593775 2035800 5852925 14307150 30045015
1 4350 40600 274050 1425060 5937750 20358000 58529250 143071500
2 1350 19575 182700 1233225 6412770 26719875 91611000 263381625
3 120 3600 52200 487200 3288600 17100720 71253000 244296000
4 210 6300 91350 852600 5755050 29926260 124692750
5 252 7560 109620 1023120 6906060 35911512
6 210 6300 91350 852600 5755050
7 120 3600 52200 487200
8 45 1350 19575
9 10 300
10 1
Total 9880 91390 658008 3838380 18643560 76904685 273438880 847660528


The next table shows the probability, to six decimal points, for each number of picks and catches.

Probability TableExpand

Catch Pick 3 Pick 4 Pick 5 Pick 6 Pick 7 Pick 8 Pick 9 Pick 10
0 0.410931 0.299869 0.216572 0.154694 0.109196 0.076106 0.052323 0.035445
1 0.440283 0.44425 0.416484 0.371266 0.318488 0.264717 0.214049 0.168784
2 0.13664 0.214192 0.277656 0.321288 0.343967 0.347441 0.335033 0.310716
3 0.012146 0.039392 0.07933 0.126929 0.176393 0.222363 0.260581 0.288200
4 0.002298 0.009574 0.023799 0.045732 0.074834 0.109444 0.147102
5 0.000383 0.00197 0.00588 0.013304 0.025256 0.042365
6 0.000055 0.000338 0.001188 0.003118 0.006789
7 0.000006 0.000047 0.000191 0.000575
8 0.000001 0.000005 0.000023
9 0.000000 0.000000
10 0.000000
Total 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000 1.000000


The next table shows the contribution to the return, to six decimal points, for each number of picks and catches. The bottom row shows the total return for each number of picks.

Return TableExpand

Catch 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
1 0 0 0 0 0 0 0 0
2 0.683198 0.428384 0.277656 0.321288 0 0 0 0
3 0.291498 0.393916 0.396652 0.253857 0.352787 0.222363 0.260581 0
4 0.142466 0.239359 0.237991 0.365853 0.299334 0.218888 0.294204
5 0.047872 0.098479 0.146994 0.10643 0.303076 0.338924
6 0.054711 0.08448 0.296959 0.077952 0.169733
7 0.006437 0.046811 0.095451 0.14369
8 0.002926 0.012343 0.028866
9 0.000366 0.003539
10 0.000024
Total 0.974696 0.964766 0.961538 0.966326 0.95655 0.974822 0.968656 0.978980


Strategy



As you can see in the table above, the greatest return is for the pick-10 at 97.90%.