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Introduction

Somebody asked me about the variance of Cleopatra Keno.I had to dust off my college statistics books to help me with that one. As a reminder, it plays like conventional spot keno, except if the last ball drawn contributes to a win, the player gets 12 free games with a 2x multiplier.

Cleopatra Keno is a video keno variant in which the player wins 12 free games if the last ball drawn contributes to a player win. All wins in free games are doubled. Free games do not earn additional free games.

As a rule, the simplest games are the card games and slots. Games like roulette and craps seem complex although they too are simple. Blackjack and baccarat are simple drawing games that even novices can play well Cleopatra Keno Non Deposit Bonus Casino in a very short space of time. Play authentic Keno games - free online Looking for Keno games that are exactly like the casino? Below are several of the most popular games available for you to play right now that you won't find anywhere else. Alien Attack Keno, Wolf Run Keno, Cleopatra Keno, and 4 Card Cleopatra Keno are GOLD Member exclusives.

Pay Tables

Following are some common pay tables for Cleopatra Keno. The return is in the bottom row.

Pay Table 1

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	-	-	-	-	-	-	-
1	0	-	-	-	-	-	-	-
2	3	1	-	-	-	-	-	-
3	19	5	3	2	1	-	-	-
4	0	45	24	5	3	3	2	1
5	0	240	42	7	10	5	3
6	0	410	118	65	12	5
7	0	500	250	108	35
8	0	1,000	200	206
9	0	1,000	1,000
10	0	2,000
Return	94.03%	95.25%	95.05%	95.03%	95.07%	95.09%	95.04%	95.02%

Pay Table 2

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	-	-	-	-	-	-	-
1	0	-	-	-	-	-	-	-
2	3	1	-	-	-	-	-	-
3	19	5	3	2	1	-	-	-
4	0	43	24	5	3	3	2	1
5	0	230	40	7	10	5	3
6	0	410	111	65	12	5
7	0	500	210	100	35
8	0	1,000	200	175
9	0	1,000	1,000
10	0	2,000
Return	94.03%	94.22%	94.17%	94.03%	94.08%	94.12%	94.19%	94.13%

Pay Table 3

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	-	-	-	-	-	-	-
1	0	-	-	-	-	-	-	-
2	3	1	-	-	-	-	-	-
3	18	5	3	2	1	-	-	-
4	0	39	23	5	3	3	2	1
5	0	225	37	7	10	5	3
6	0	390	97	60	10	5
7	0	500	200	100	30
8	0	1,000	200	160
9	0	1,000	1,000
10	0	2,000
Return	92.11%	92.16%	92.08%	92.10%	92.09%	92.08%	92.14%	92.00%

Pay Table 4

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	-	-	-	-	-	-	-
1	0	-	-	-	-	-	-	-
2	3	1	-	-	-	-	-	-
3	17	5	3	2	1	-	-	-
4	0	35	22	5	3	3	2	1
5	0	220	34	6	10	5	3
6	0	375	95	56	10	5
7	0	500	180	80	25
8	0	1,000	200	150
9	0	1,000	1,000
10	0	2,000
Return	90.19%	90.10%	89.99%	90.28%	90.14%	90.15%	90.02%	90.01%

Pay Table 5

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0	-	-	-	-	-	-	-
1	0	-	-	-	-	-	-	-
2	3	1	-	-	-	-	-	-
3	16	5	3	2	1	-	-	-
4	0	31	21	5	3	3	2	1
5	0	220	30	6	10	5	3
6	0	375	80	50	10	5
7	0	500	180	65	25
8	0	1,000	200	100
9	0	1,000	1,000
10	0	2,000
Return	88.27%	88.04%	88.34%	88.27%	88.01%	87.99%	88.44%	88.58%

Analysis

This section shall analyze Pay Table 4 above.

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

Cleopatra Keno — CombinationsExpand

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	34220	487635	5461512	50063860	386206920	2558620845	14783142660	75394027566
1	35400	684400	9752700	109230240	1001277200	7724138400	51172416900	295662853200
2	11400	336300	6501800	92650650	1037687280	9512133400	73379314800	486137960550
3	1140	68400	2017800	39010800	555903900	6226123680	57072800400	440275888800
4	0	4845	290700	8575650	165795900	2362591575	26461025640	242559401700
5	0	0	15504	930240	27442080	530546880	7560293040	84675282048
6	0	0	0	38760	2325600	68605200	1326367200	18900732600
7	0	0	0	0	77520	4651200	137210400	2652734400
8	0	0	0	0	0	125970	7558200	222966900
9	0	0	0	0	0	0	167960	10077600
10	0	0	0	0	0	0	0	184756
Total	82160	1581580	24040016	300500200	3176716400	28987537150	231900297200	1646492110120

The next table shows the probability for each possible number of picks and catches, before considering the bonus.

Cleopatra Keno — Probability of WinningExpand

Catch	Pick 3	Pick 4	Pick 5	Pick 6	Pick 7	Pick 8	Pick 9	Pick 10
0	0.41650438	0.30832143	0.22718421	0.16660175	0.12157425	0.08826624	0.06374784	0.0457907
1	0.4308666	0.43273183	0.40568609	0.36349473	0.31519251	0.26646411	0.22066559	0.17957138
2	0.13875365	0.21263547	0.27045739	0.30832143	0.32665405	0.32814562	0.31642614	0.29525678
3	0.01387537	0.04324789	0.08393505	0.12981955	0.17499324	0.21478623	0.24610922	0.26740237
4	0	0.00306339	0.01209234	0.02853792	0.05219097	0.0815037	0.11410518	0.1473189
5	0	0	0.00064492	0.00309564	0.0086385	0.01830259	0.03260148	0.05142769
6	0	0	0	0.00012898	0.00073208	0.00236671	0.00571956	0.01147939
7	0	0	0	0	0.0000244	0.00016046	0.00059168	0.00161114
8	0	0	0	0	0	0.00000435	0.00003259	0.00013542
9	0	0	0	0	0	0	0.00000072	0.00000612
10	0	0	0	0	0	0	0	0.00000011
Total	1	1	1	1	1	1	1	1

The next table shows the contribution to the return for each possible number of picks and catches, before considering the bonus.

Cleopatra Keno — Before Bonus ReturnExpand

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	0.41626095	0.21263547	0	0	0	0	0	0
3	0.23588121	0.21623946	0.25180516	0.2596391	0.17499324	0	0	0
4	0	0.10721873	0.26603144	0.14268959	0.1565729	0.2445111	0.22821036	0.1473189
5	0	0	0.14188343	0.10525171	0.05183103	0.18302586	0.1630074	0.15428306
6	0	0	0	0.04836935	0.06954728	0.13253596	0.05719558	0.05739697
7	0	0	0	0	0.01220128	0.02888193	0.04733427	0.04027858
8	0	0	0	0	0	0.00434566	0.00651849	0.0203129
9	0	0	0	0	0	0	0.00072428	0.00612065
10	0	0	0	0	0	0	0	0.00022442
Total	0.65214216	0.53609365	0.65972003	0.55594975	0.46514573	0.59330052	0.50299039	0.42593549

The next table shows the probability of winning the bonus for each possible number of picks and catches, before considering the bonus.

Cleopatra Keno — Bonus ProbabilityExpand

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	0.01387537	0.02126355	0	0	0	0	0	0
3	0.0020813	0.00648718	0.01259026	0.01947293	0.02624899	0	0	0
4	0	0.00061268	0.00241847	0.00570758	0.01043819	0.01630074	0.02282104	0.02946378
5	0	0	0.00016123	0.00077391	0.00215963	0.00457565	0.00815037	0.01285692
6	0	0	0	0.0000387	0.00021962	0.00071001	0.00171587	0.00344382
7	0	0	0	0	0.00000854	0.00005616	0.00020709	0.0005639
8	0	0	0	0	0	0.00000174	0.00001304	0.00005417
9	0	0	0	0	0	0	0.00000033	0.00000275
10	0	0	0	0	0	0	0	0.00000006
Total	0.01595667	0.02836341	0.01516996	0.02599312	0.03907497	0.0216443	0.03290772	0.0463854

The next table shows the return for each combinations of picks and catches, according to Pay Table 1 above.

Cleopatra Keno — Return

Pick	Return Before Bonus	Bonus Return	Total Return
3	0.652142	0.249744	0.901887
4	0.536094	0.364931	0.901024
5	0.659720	0.240190	0.899910
6	0.555950	0.346821	0.902771
7	0.465146	0.436213	0.901359
8	0.593301	0.308198	0.901498
9	0.502990	0.397254	0.900245
10	0.425935	0.474172	0.900108

So, the bottom line is that this particular pay table returns 90% of money bet.

Calculator

Calculate the odds for any pay table with my Cleopatra Keno calculator.

Written by:Michael Shackleford

Thread Rating:

Wizard
Administrator
Somebody asked me about the variance of Cleopatra Keno. I had to dust off my college statistics books to help me with that one. As a reminder, it plays like conventional spot keno, except if the last ball drawn contributes to a win, the player gets 12 free games with a 2x multiplier. Free games do not earn more free games. Let's use the 3-10-56-180-1000 pick-8 pay table as an example.
To begin, recall var(x + y) = var(x) + var(y) + 2*cov(x,y)
In this case of this game, var(entire game) = var(base game) + var(bonus) + 2*cov(base game,bonus)
Let's start with the variance of the base game.

Catch	Pays	Winning combinations	Probability	Return	exp win^2
0	0	2,558,620,845	0.088266	0.000000	0.000000
1	0	7,724,138,400	0.266464	0.000000	0.000000
2	0	9,512,133,400	0.328146	0.000000	0.000000
3	0	6,226,123,680	0.214786	0.000000	0.000000
4	3	2,362,591,575	0.081504	0.244511	0.733533
5	10	530,546,880	0.018303	0.183026	1.830259
6	56	68,605,200	0.002367	0.132536	7.422014
7	180	4,651,200	0.000160	0.028882	5.198747
8	1000	125,970	0.000004	0.004346	4.345661
Total	28,987,537,150	1.000000	0.593301	19.530214

As a reminder, the variance equals exp(x^2) - (e(x))^2.
So, var(base game) = 19.530214 - 0.593301^2 = 19.178208.
Next, let's do the variance of the bonus, given that the player won the bonus in the first place. Recall var(ax) = a^2 * var(x). In the bonus there are 12 doubled free games. So the variance of a bonus would be...
12 * 2^2 * 19.178208 = 920.554000.
However, the player doesn't always win the bonus. This was a tricky step. It would have been nice to just multiply that by the probability of winning the bonus of 0.021644, but you can't.
Let's do the easy part first. The average bonus win is 12*2*0.593301 = 14.239212. The expected win from the bonus on any given spin is prob(bonus)*(average bonus) = 0.021644 * 14.23921236 = 0.308198. Not that we directly care, but the overall return of the game is exp(base game) + exp(bonus) = 0.593301 + 0.308198 = 0.901498.
To find the exp(x^2) of the bonus, recall variance = exp(x^2) - (e(x))^2.
To rearrange:
exp(x^2) = var(x) + (e(x))^2.
In the case of the bonus, given a bonus win in the first place:
exp(x^2) = 920.554000 + 14.239212^2 = 1123.309169.
Now we're ready to find the overall variance from the bonus (including when the player doesn't win it):
We already know exp(bonus win) = 0.308198.
exp((bonus win)^2) = 0.021644 * 1123.309169 = 24.313239.
Thus the variance of the bonus on each spin is 24.313239 - 0.308198^2 = 24.218253.
Next, let's do the covariance. Why is there any covariance, you might ask? It's because the player has to hit a winning ball on the last draw to trigger the bonus. Given the fact that the last ball won makes it more likely the player won money in the base game on that spin. In other words, winning the bonus is correlated to winning something on the base spin.
We'll need to know the expected win in the base game, given that the bonus was won. Here is that table:

Catch	Pays	Winning combinations	Probability	Return
0	0	-	0.000000	0.000000
1	0	-	0.000000	0.000000
2	0	-	0.000000	0.000000
3	0	-	0.000000	0.000000
4	3	472,518,315	0.753119	2.259358
5	10	132,636,720	0.211402	2.114019
6	56	20,581,560	0.032804	1.837010
7	180	1,627,920	0.002595	0.467036
8	1000	50,388	0.000080	0.080310
Total	627,414,903	1.000000	6.757734

From our college statistics class, we know cov(x,y) = exp(xy) - exp(x)*exp(y).
Exp(xy) = 0.021644 * 6.757734 * 14.239212 = 2.082719.
So, the overall covariance is 2.082719 - 0.593301 * 0.308198 = 1.899865.
Thus, our bottom line is:
var(base game) + var(bonus) + 2*cov(base game,bonus) = 19.178208 + 24.218253 - 2*1.899865 = 47.19619.
Here is a summary of the key numbers.

Exp base win	0.593301
Exp bonus	0.308198
Return of game	0.901498
Average base win given bonus won	6.757734
Average bonus	14.239212
Prob bonus	0.021644
Variance base win	19.178208
Variance bonus, given bonus	920.554000
win^2 in bonus	1123.309169
Variance bonus	24.218253
covariance	1.899865
total variance	47.196191

I hope the one who asked me (I'm not sure if I can state his name) is happy, this took hours, including running a simulation to confirm the answer.As usual, a post that took me hours to write gets no replies.ck1313

As usual, a post that took me hours to write gets no replies.

Come on Mike, it's only been 3 hours since you posted it. :P
And well done! I haven't done much work with covariance. I took college statistics, but I definitely didn't learn covariance well and definitely hadn't seen it applied to a gaming example with careful detail. This is a fairly difficult calculation for most adults to do, and I think you guided the reader very well. So if even the person who asked didn't learn anything, at least I did!

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Also writing any post on here that uses a table is a royal PITA and will take forever to write unless you have a nice script to add rows and columns.Thanks guys for the replies, although I'm sure they were sympathetic. To be honest, it was good for me to review those formulas and I'll get mileage from it as a future 'ask the wizard' question.Gialmere

As usual, a post that took me hours to write gets no replies.

Sorry. I hang out at a website that taught me Keno was for suckers so I tend not to read articles about the game.djatcmiplet

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Also writing any post on here that uses a table is a royal PITA and will take forever to write unless you have a nice script to add rows and columns.

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I've been using http://miplet.net/table/ for a long time. Just copy and paste from a spreadsheet.

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