Keno Odds
Keno is an entirely electronic game that is played every few minutes by a computer. Pretend that the computer has a basket of eighty balls, each sequentially numbered from one to eighty. You take a card with numbers from one to eighty and mark from one to fifteen “spots” (i.e., “lucky balls”) that you expect to be drawn, and then place a bet. After the bets are closed, the computer pulls out twenty balls from the basket, and if you “catch” enough “spots,” you’ll get a return on your bet.
Summary of Results
If following section’s calculations are correct, along with my program, the fraction of your bet that you lose on average per game for this particular casino depends on the number of spots that you play on the card, as follows:
| Spots bet | Percentage lost |
|---|---|
| 1 | 25% |
| 4 | 26% |
| 3 | 26% |
| 5 | 27% |
| 8 | 27% |
| 7 | 28% |
| 10 | 28% |
| 2 | 28% |
| 11 | 28% |
| 9 | 28% |
| 12 | 28% |
| 6 | 28% |
| 13 | 29% |
| 14 | 29% |
| 15 | 29% |
So, from what I can tell, you’re crazy to play Keno for any reason. If you can get your hands on the odds brochure from your favorite casino (or state lottery) and email them to me, I’ll be happy to run my program for you, or you can try it yourself. The theory is so simple that you can write your own program in a few minutes, and even if you cannot program, all you need is a calculator to determine the odds yourself.
How to Calculate the Odds for your Casino.
The probability of catching exactly r spots when you bet on N of them (where N >= r) is given by
P(N,r) = c(N,r) * c(80-N, 20-r) / c(80,20).
In other words, P(N,r) is the number of ways that you can pick r of the N spots times the number of ways that the computer can pick all of the spots that you didn’t bet on divided by the ways that it can pick twenty spots. Note that c(N,r) is the famous “binomial coefficient”, where
c(N,r) = f(N,r) / f(r,r),
and we can define f(n,r) according to the rules
f(n,0) = 1, otherwise f(n,r) = n * f(n-1,r-1).
Example
A casino says “play four spots, catch two and get 1:1, three and get 4:1, all four and get 115:1.” Should we bet on four spots?
The probability of getting zero of four spots is p0 = P(4,0) = 97527 / 316316 = 0.3083. The probability of getting one of four spots is p1 = P(4,1)= 34220 / 79079 = 0.4327. The probability of getting two of four spots is p2 = P(4, 2) = 16815 / 79079 = 0.2126. The probability of getting three of four spots is p3 = P(4, 3) = 3420 / 79079 = 0.04324. The probability of getting all four spots is p4 = P(4, 4) = 969 / 316316 = 0.003063. (Note that the values of P can be found in the following table for your convenience.)
Note that all of these probabilities add to one: p0 + p1 + p2 + p3 + p4 = 1. Now you start by giving them one dollar, but you have a chance to win it back! You earn $1 with probability p2, $4 with probability p3, and $115 with probability p4, so your expected return per dollar bet is
=(what you put in) + (what you expect to get out)
= (-$1) + (($1)p2 + ($4)p3 + ($115)p4)
= (-$1) + (($0.213) + ($0.173) + (0.35229012))
= (-$1) + ($0.738)
= -$0.26.
In other words, you expect to loose about 0.26 cents per dollar that you bet on four spots, and this is horrible—pick a better game, like Baccarat.
A Specific Casino
For a dollar bet,
| Spots | Catch | Win | Probability | Expected return |
|---|---|---|---|---|
| 1 | 1 | $3.00 | 0.25 | $0.7500 |
| 2 | 2 | $12.00 | 0.060126584 | $0.7215 |
| 3 | 2 | $1.00 | 0.13875365 | $0.1388 |
| 3 | 3 | $43.00 | 0.013875365 | $0.5966 |
| 4 | 2 | $1.00 | 0.21263547 | $0.2126 |
| 4 | 3 | $4.00 | 0.04324789 | $0.1730 |
| 4 | 4 | $115.00 | 0.0030633924 | $0.3523 |
| 5 | 3 | $2.00 | 0.08393505 | $0.1679 |
| 5 | 4 | $20.00 | 0.012092338 | $0.2418 |
| 5 | 5 | $500.00 | 6.449247e-4 | $0.3225 |
| 6 | 3 | $1.00 | 0.12981954 | $0.1298 |
| 6 | 4 | $4.00 | 0.028537918 | $0.1142 |
| 6 | 5 | $90.00 | 0.0030956385 | $0.2786 |
| 6 | 6 | $1500.00 | 1.2898494e-4 | $0.1935 |
| 7 | 3 | $0.50 | 0.17499325 | $0.0875 |
| 7 | 4 | $1.50 | 0.052190967 | $0.0783 |
| 7 | 5 | $20.00 | 0.008638505 | $0.1728 |
| 7 | 6 | $360.00 | 7.320767e-4 | $0.2635 |
| 7 | 7 | $5000.00 | 2.4402556e-5 | $0.1220 |
| 8 | 5 | $9.00 | 0.018302586 | $0.1647 |
| 8 | 6 | $90.00 | 0.0023667137 | $0.2130 |
| 8 | 7 | $1500.00 | 1.6045517e-4 | $0.2407 |
| 8 | 8 | $25000.00 | 4.3456605e-6 | $0.1086 |
| 9 | 4 | $0.50 | 0.11410519 | $0.0571 |
| 9 | 5 | $3.00 | 0.03260148 | $0.0978 |
| 9 | 6 | $40.00 | 0.0057195583 | $0.2288 |
| 9 | 7 | $300.00 | 5.9167844e-4 | $0.1775 |
| 9 | 8 | $4000.00 | 3.2592455e-5 | $0.1304 |
| 9 | 9 | $37500.00 | 7.242768e-7 | $0.0272 |
| 10 | 5 | $2.00 | 0.05142769 | $0.1029 |
| 10 | 6 | $20.00 | 0.0114793945 | $0.2296 |
| 10 | 7 | $140.00 | 0.0016111432 | $0.2256 |
| 10 | 8 | $1000.00 | 1.3541937e-4 | $0.1354 |
| 10 | 9 | $4000.00 | 6.120649e-6 | $0.0245 |
| 10 | 10 | $50000.00 | 1.1221189e-7 | $0.0056 |
| 11 | 5 | $1.00 | 0.074080355 | $0.0741 |
| 11 | 6 | $8.00 | 0.020203736 | $0.1616 |
| 11 | 7 | $80.00 | 0.0036078098 | $0.2886 |
| 11 | 8 | $315.00 | 4.1141692e-4 | $0.1296 |
| 11 | 9 | $1800.00 | 2.837358e-5 | $0.0511 |
| 11 | 10 | $12500.00 | 1.05799794e-6 | $0.0132 |
| 11 | 11 | $65000.00 | 1.603027e-8 | $0.0010 |
| 12 | 5 | $0.50 | 0.09938732 | $0.0497 |
| 12 | 6 | $3.00 | 0.032208852 | $0.0966 |
| 12 | 7 | $35.00 | 0.0070273858 | $0.2460 |
| 12 | 8 | $260.00 | 0.0010195985 | $0.2651 |
| 12 | 9 | $500.00 | 9.5401025e-5 | $0.0477 |
| 12 | 10 | $1500.00 | 5.427989e-6 | $0.0081 |
| 12 | 11 | $20000.00 | 1.672724e-7 | $0.0033 |
| 12 | 12 | $70000.00 | 2.0909049e-9 | $0.0001 |
| 13 | 0 | $1.00 | 0.016395647 | $0.0164 |
| 13 | 6 | $1.00 | 0.047501296 | $0.0475 |
| 13 | 7 | $18.00 | 0.012315149 | $0.2217 |
| 13 | 8 | $80.00 | 0.0021831403 | $0.1747 |
| 13 | 9 | $700.00 | 2.5989765e-4 | $0.1819 |
| 13 | 10 | $3000.00 | 2.0062272e-5 | $0.0602 |
| 13 | 11 | $10000.00 | 9.433671e-7 | $0.0094 |
| 13 | 12 | $50000.00 | 2.3983909e-8 | $0.0012 |
| 13 | 13 | $75000.00 | 2.459888e-10 | $0.0000 |
| 14 | 0 | $1.00 | 0.011501424 | $0.0115 |
| 14 | 6 | $1.00 | 0.06575738 | $0.0658 |
| 14 | 7 | $10.00 | 0.019851286 | $0.1985 |
| 14 | 8 | $40.00 | 0.0041816365 | $0.1673 |
| 14 | 9 | $310.00 | 6.0823804e-4 | $0.1886 |
| 14 | 10 | $1100.00 | 5.973766e-5 | $0.0657 |
| 14 | 11 | $3100.00 | 3.8110152e-6 | $0.0118 |
| 14 | 12 | $25000.00 | 1.4784111e-7 | $0.0037 |
| 14 | 13 | $50000.00 | 3.084039e-9 | $0.0002 |
| 14 | 14 | $100000.00 | 2.5700322e-11 | $0.0000 |
| 15 | 0 | $1.00 | 0.008016144 | $0.0080 |
| 15 | 6 | $1.00 | 0.08634808 | $0.0863 |
| 15 | 7 | $5.00 | 0.02988972 | $0.1494 |
| 15 | 8 | $30.00 | 0.0073314407 | $0.2199 |
| 15 | 9 | $130.00 | 0.0012671626 | $0.1647 |
| 15 | 10 | $310.00 | 1.5205951e-4 | $0.0471 |
| 15 | 11 | $2500.00 | 1.2342492e-5 | $0.0309 |
| 15 | 12 | $7500.00 | 6.4960486e-7 | $0.0049 |
| 15 | 13 | $25000.00 | 2.0677078e-8 | $0.0005 |
| 15 | 14 | $50000.00 | 3.5045895e-10 | $0.0000 |
| 15 | 15 | $125000.00 | 2.336393e-12 | $0.0000 |
Keno Odds
Wherever you choose to play the payout schedule overall is geared to provide around a 70% return for live Keno and around 85-90% for Online and Video Keno. This doesn’t sem like a big difference but think of it in the inverse terms - the casino’s profit. A game that gives you a 95% return means the casino is keeping 5% but a game that is giving you a 70% return then the casino is keeping 30% - 6 times as much! You can download my Keno Odds spreadsheet by right-clicking KenoOdds.xls and selecting “Save target as” to figure out what the casino edge is in your keno game.
The equation for calculating the probability (p) of hitting (n) numbers out of the (x) numbers you picked when (y) numbers were drawn out of (z). (i.e. - “What is the probability of hitting 4 out of the 5 numbers I picked when 20 numbers were drawn out of 80″).
n = 4
x = 5
y = 20
z = 80
p(x,n) = (combin(x,n) * combin(z-x,y-x+n)) / combin(z,y)
It is important not to look at the payout schedules but to actually see how much profit the casino is keep from the money you wager. That’s what the following does. I took 2 payout schedules and calculated the casino edge on all the bets and showed it in red. At first glance the 2nd payout schedule looks better because the high-end payouts are much higher but payouts are smaller on the lower end. But the payouts on the small end occur much more frequently and that’s where the value of the ticket is most of the time. The 1st payout schedule shows a casino edge of about 7% while the 2nd schedule shows an edge anywhere 21-66%! You will lose your money anywhere from 3-9 times as fast playing the 2nd schedule if you choose to play the game with the big payoffs.
Keno Odds
Here are all possible keno outcomes. For every possible number of spots played you can see both the probability of each outcome and the odds against that outcome happening. This shows, for example, that you have roughly a 71-1 shot at hitting a 3-spot, while you are almost 85% likely to catch only 0 or 1 on the same ticket. For very large or very small numbers, scientific notation is used. The term “e+nnn” means move the decimal point nnn places to the right, adding zeroes as appropriate. Similarly, “e-nnn” means to move the decimal point nnn places to the left, padding with zeroes. Thus, 1.42e-09 is shorthand for .00000000142, while 2.71e+6 is another way of writing 2710000.
Keno outcome probabilities
Play 1 spot
Catch probability odds-to-1 against
0 0.7500 0.3333
1 0.2500 3.0000
Play 2 spots
Catch probability odds-to-1 against
0 0.5601 0.7853
1 0.3797 1.6333
2 0.0601 15.6316
Play 3 spots
Catch probability odds-to-1 against
0 0.4165 1.4009
1 0.4309 1.3209
2 0.1388 6.2070
3 0.0139 71.0702
Play 4 spots
Catch probability odds-to-1 against
0 0.3083 2.2434
1 0.4327 1.3109
2 0.2126 3.7029
3 0.0432 22.1225
4 0.003063 325.44
Play 5 spots
Catch probability odds-to-1 against
0 0.2272 3.4017
1 0.4057 1.4650
2 0.2705 2.6974
3 0.0839 10.9140
4 0.0121 81.6970
5 0.000645 1549.57
Play 6 spots
Catch probability odds-to-1 against
0 0.1666 5.0023
1 0.3635 1.7511
2 0.3083 2.2434
3 0.1298 6.7030
4 0.0285 34.0411
5 0.003096 322.04
6 0.000129 7751.84
Play 7 spots
Catch probability odds-to-1 against
0 0.1216 7.2254
1 0.3152 2.1727
2 0.3267 2.0613
3 0.1750 4.7145
4 0.0522 18.1604
5 0.008639 114.76
6 0.000732 1364.98
7 0.00002440 40978
Play 8 spots
Catch probability odds-to-1 against
0 0.0883 10.3294
1 0.2665 2.7529
2 0.3281 2.0474
3 0.2148 3.6558
4 0.0815 11.2694
5 0.0183 53.6371
6 0.002367 421.53
7 0.000160 6231.27
8 0.00000435 230114
Play 9 spots
Catch probability odds-to-1 against
0 0.0637 14.6868
1 0.2207 3.5317
2 0.3164 2.1603
3 0.2461 3.0632
4 0.1141 7.7638
5 0.0326 29.6735
6 0.005720 173.84
7 0.000592 1689.11
8 0.00003259 30681
9 7.2428e-007 1.3807e+006
Play 10 spots
Catch probability odds-to-1 against
0 0.0458 20.8385
1 0.1796 4.5688
2 0.2953 2.3869
3 0.2674 2.7397
4 0.1473 5.7880
5 0.0514 18.4448
6 0.0115 86.1126
7 0.001611 619.68
8 0.000135 7383.47
9 0.00000612 163380
10 1.1221e-007 8.9117e+006
Play 11 spots
Catch probability odds-to-1 against
0 0.0327 29.5739
1 0.1439 5.9486
2 0.2681 2.7303
3 0.2784 2.5921
4 0.1786 4.5995
5 0.0741 12.4989
6 0.0202 48.4958
7 0.003608 276.18
8 0.000411 2429.62
9 0.00002837 35243
10 0.00000106 945180
11 1.6030e-008 6.2382e+007
Play 12 spots
Catch probability odds-to-1 against
0 0.0232 42.0530
1 0.1138 7.7900
2 0.2378 3.2057
3 0.2797 2.5749
4 0.2058 3.8600
5 0.0994 9.0616
6 0.0322 30.0474
7 0.007027 141.30
8 0.001020 979.78
9 0.00009540 10481
10 0.00000543 184229
11 1.6727e-007 5.9783e+006
12 2.0909e-009 4.7826e+008
Play 13 spots
Catch probability odds-to-1 against
0 0.0164 59.9918
1 0.0888 10.2600
2 0.2066 3.8398
3 0.2727 2.6665
4 0.2273 3.3998
5 0.1259 6.9442
6 0.0475 20.0521
7 0.0123 80.2008
8 0.002183 457.06
9 0.000260 3846.67
10 0.00002006 49844
11 9.4337e-007 1.0600e+006
12 2.3984e-008 4.1695e+007
13 2.4599e-010 4.0652e+009
Play 14 spots
Catch probability odds-to-1 against
0 0.0115 85.9458
1 0.0685 13.5945
2 0.1763 4.6723
3 0.2590 2.8603
4 0.2422 3.1287
5 0.1520 5.5801
6 0.0658 14.2074
7 0.0199 49.3746
8 0.004182 238.14
9 0.000608 1643.09
10 0.00005974 16739
11 0.00000381 262396
12 1.4784e-007 6.7640e+006
13 3.0840e-009 3.2425e+008
14 2.5700e-011 3.8910e+010
Play 15 spots
Catch probability odds-to-1 against
0 0.008016 123.75
1 0.0523 18.1281
2 0.1479 5.7595
3 0.2404 3.1597
4 0.2502 2.9966
5 0.1762 4.6770
6 0.0863 10.5810
7 0.0299 32.4563
8 0.007331 135.40
9 0.001267 788.16
10 0.000152 6575.37
11 0.00001234 81020
12 6.4960e-007 1.5394e+006
13 2.0677e-008 4.8363e+007
14 3.5046e-010 2.8534e+009
15 2.3364e-012 4.2801e+011
Play 16 spots
Catch probability odds-to-1 against
0 0.005550 179.19
1 0.0395 24.3395
2 0.1223 7.1798
3 0.2185 3.5768
4 0.2515 2.9762
5 0.1971 4.0738
6 0.1084 8.2251
7 0.0425 22.5240
8 0.0120 82.6408
9 0.002406 414.59
10 0.000343 2913.53
11 0.00003403 29387
12 0.00000228 438862
13 9.8402e-008 1.0162e+007
14 2.5449e-009 3.9295e+008
15 3.4507e-011 2.8980e+010
16 1.7972e-013 5.5641e+012
Play 17 spots
Catch probability odds-to-1 against
0 0.003815 261.10
1 0.0295 32.9185
2 0.0996 9.0417
3 0.1948 4.1324
4 0.2467 3.0542
5 0.2138 3.6779
6 0.1309 6.6405
7 0.0576 16.3649
8 0.0183 53.4990
9 0.004234 235.16
10 0.000703 1421.34
11 0.00008285 12069
12 0.00000678 147516
13 3.7247e-007 2.6848e+006
14 1.3069e-008 7.6517e+007
15 2.7039e-010 3.6983e+009
16 2.8643e-012 3.4912e+011
17 1.1233e-014 8.9026e+013
Play 18 spots
Catch probability odds-to-1 against
0 0.002604 383.00
1 0.0218 44.8670
2 0.0800 11.4963
3 0.1707 4.8576
4 0.2366 3.2267
5 0.2255 3.4342
6 0.1527 5.5490
7 0.0748 12.3710
8 0.0267 36.4013
9 0.006990 142.06
10 0.001331 750.43
11 0.000183 5475.01
12 0.00001775 56324
13 0.00000119 839002
14 5.3209e-008 1.8794e+007
15 1.4936e-009 6.6952e+008
16 2.4142e-011 4.1421e+010
17 1.9256e-013 5.1932e+012
18 5.3489e-016 1.8695e+015
Play 19 spots
Catch probability odds-to-1 against
0 0.001764 565.86
1 0.0160 61.6531
2 0.0635 14.7549
3 0.1471 5.7962
4 0.2223 3.4975
5 0.2320 3.3101
6 0.1728 4.7879
7 0.0936 9.6853
8 0.0372 25.8502
9 0.0109 90.5347
10 0.002356 423.39
11 0.000371 2696.22
12 0.00004197 23824
13 0.00000335 298668
14 1.8263e-007 5.4756e+006
15 6.5224e-009 1.5332e+008
16 1.4304e-010 6.9912e+009
17 1.7408e-012 5.7445e+011
18 9.8350e-015 1.0168e+014
19 1.7254e-017 5.7956e+016
Play 20 spots
Catch probability odds-to-1 against
0 0.001186 842.38
1 0.0116 85.4464
2 0.0497 19.1150
3 0.1249 7.0087
4 0.2050 3.8773
5 0.2333 3.2867
6 0.1902 4.2583
7 0.1133 7.8265
8 0.0499 19.0554
9 0.0163 60.4198
10 0.003940 252.80
11 0.000702 1422.82
12 0.00009117 10968
13 0.00000847 118084
14 5.4888e-007 1.8219e+006
15 2.3951e-008 4.1751e+007
16 6.6828e-010 1.4964e+009
17 1.1035e-011 9.0624e+010
18 9.5126e-014 1.0512e+013
19 3.3943e-016 2.9461e+015
20 2.8286e-019 3.5353e+018