User:JohnH~enwiki/sandboxP

Reasons for this page

This page exists for the purposes of archiving my major contributions to Wikipedia. Someone may delete my efforts and I would like to capture them for future use. Archives so far are:

• Keno odds
• Hypergeometric distribution comments

Keno

Background

Keno payouts are based on how many numbers the player chooses and how many of those numbers are hit, multiplied by the proportion of the player's original wager to the base rate of the house's paytable. Typically, the more numbers a player chooses and the more numbers hit, the greater the payout, although some paytables pay for hitting a lesser number of spots. For example, it is not uncommon to see casinos paying $500 or even $1,000 for a catch of 0 out of 20 on a 20-spot ticket with a $5.00 wager. Payouts vary widely by casino. Most casinos allow paytable wagers of 1 through 20 numbers, but some limit the choice to only 1 through 10, 12 and 15 numbers, or spots as keno aficionados call the numbers selected.

The probability of a player hitting all 20 numbers on a 20-spot ticket is approximately 1 in 3.5 quintillion (1 in 3,535,316,142,212,174,320).

Even though it is highly improbable to hit all 20 numbers on a 20 spot ticket, the same player would typically also get paid for catching 0, 1, 2, 3, and 7 through 19 out of 20, often with the 17 through 19 catches paying the same as the solid 20 hit. Some of the other paying catches on a 20-spot ticket or any other ticket with high solid catch odds are in reality very possible to hit.

Below is a set of tables giving both the probability and odds of catching N numbers when playing M spots for all 0 ≤ N ≤ M ≤ 20 where M ≥1.

This assumes a “casino standard” of drawing 20 balls from a group of 80.

Note there is a slight (but important) difference between the odds to 1 figure (given below) and a similar value, the 1 chance in figure (not given). Consider the simple case of catching exactly 1 number when playing a 1-spot ticket. While the odds figure reads “3 to 1” against, the chance in figure is “1 in 4”. The tables below give only the odds to 1 figure, since it's quite easy to calculate the chance in figure: just add 1 to the odds figure to get the chance in figure as was done in the 3-to-1 example above. When the odds against figure gets large (as it often does in Keno!) you can consider the two to be interchangeable for all intents and purposes.

Similarly, if you prefer your probabilities in percent form, simply take the probability number and move the decimal point two places to the RIGHT, drop any leading 0's (those to the LEFT of the decimal point) and stick a %-sign on the end of the value. Thus the probability of catching 2 numbers on a 3-spot ticket reads 0.1388 or (moving the decimal point 2 places) 013.88% or just 13.88%, for example.

For very large, or very small (but positive) values the numbers are shown in “e-notation” where values are displayed as x.xxxxe±nnn. In these cases the e should be read as “times 10 to the power of” nnn, or “×10^nnn”. For example, the odds against (not probability of) catching 13 spots on a 14-spot ticket are 3.2425e+008 to 1. This is 3.2425 × 10^+008. (+008 is of course just 8). You can convert to full decimal notation by moving the decimal point 8 places to the RIGHT (because of the '+' in e+008), supplying 0's for placeholders as needed. In this case 3.2425e+008 is (just adding extra 0's to fill) 324250000. to 1 against. This could also be written as 324,250,000. to 1 against. (In the USA anyhow. In Europe the uses of ',' and '.' in numbers are interchanged.)

For negative values of nnn you do the same thing but shift the decimal point to the LEFT. For example, the probability (not odds) of catching 13 spots on a 14-spot ticket is 3.0840e−009. This is 3.0840 × 10^−009 (−009 is of course just −9). You can convert to decimal notation by moving the decimal point 9 places to the LEFT (because of the '−' in e−009), supplying 0's for placeholders as needed. In this case 3.0840e−009 is (just adding extra 0's to fill) .0000000030840 or sometimes .000 000 003 084 by analogy with the use of commas for large numbers.

Probability Tables

The following tables should be self-explanatory.

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

Derivation of Tables

Keno probabilities come from a Hypergeometric distribution, specifically its Application to Keno. To calculate the probability of hitting 4 spots on a 6-spot ticket, the formula is:

$P(X=4)={{{6 \choose 4}{{80-6} \choose {20-4}}} \over {80 \choose 20}}\approx 0.02853791$

where ${n \choose r}$ is calculated as $n! \over r!(n-r)!$, where X! is notation for X Factorial.

Some calculators (e.g. the TI-36X) and spreadsheets (e.g., Microsoft Excel) have built-in functions for ${n \choose r}$. In Microsoft Excel the function is COMBIN(n,r). Entering =COMBIN(8,5)to Excel results in the number 56 appearing in the cell.

For Keno one calculates the probability of hitting exactly $r$ spots on an $n$-spot ticket by the formula:

P(hitting $r$ spots) $={{n \choose r}\times {{80-n} \choose {20-r}} \over {80 \choose 20}}$ for an $n$-spot ticket.

The probability of hitting exactly $8$ spots on a $14$-spot ticket is therefore ${{{14 \choose 8}\times {66 \choose 12}} \over {80 \choose 20}}$. This is ${{3003\times (4.922879482\times 10^{12})} \over 3.535316142\times 10^{18}}$ $\approx 0.004181637$.

Hypergeometric Distribution

Application to Keno

The hypergeometric distribution is indispensable for calculating Keno odds. In Keno, 20 balls are randomly drawn from a collection of 80 numbered balls in a container, rather like American Bingo. Prior to each draw, a player selects a certain number of spots by marking a paper form supplied for this purpose. For example, a player might play a 6-spot by marking 6 numbers, each from a range of 1 through 80 inclusive. Then (after all players have taken their forms to a cashier and been given a duplicate of their marked form, and paid their wager) 20 balls are drawn. Some of the balls drawn may match some or all of the balls selected by the player. Generally speaking, the more hits (balls drawn that match player numbers selected) the greater the payoff.

For example, if a customer bets (“plays”) $1 for a 6-spot (not an uncommon example) and hits 4 out of the 6, the casino would pay out $4. Payouts can vary from one casino to the next, but $4 is a typical value here. The probability of this event is:

$P(X=4)=f(4;80,6,20)={{{6 \choose 4}{{80-6} \choose {20-4}}} \over {80 \choose 20}}\approx 0.02853791$

Similarly, the chance for hitting 5 spots out of 6 selected is ${{{6 \choose 5}{{74} \choose {15}}} \over {80 \choose 20}}\approx 0.003095639$ while a typical payout might be $88. The payout for hitting all 6 would be around $1500 (probability ≈ 0.000128985 or 7752-to-1). The only other nonzero payout might be $1 for hitting 3 numbers (i.e., you get your bet back), which has a probability near 0.129819548.

Taking the sum of products of payouts times corresponding probabilities we get an expected return of 0.70986492 or roughly 71% for a 6-spot, for a house advantage of 29%. Other spots-played have a similar expected return. This very poor return (for the player) is usually explained by the large overhead (floor space, equipment, personnel) required for the game.

A complete set of Keno probabilities can be found in the Wikipedia Keno article. A fairly complete set of typical payouts can be found in Probabilities in Keno. Other payouts can be found by searching the web for “keno payouts”, but formats vary and are less convenient than the site referenced.