Definitions

# Keno

[kee-noh]
This article describes the lottery game. For the radio station, see KENO. For the nuclear criticality code, see Keno (computer program).

Keno is a lottery-like or bingo-like gambling game played at modern casinos. A traditional live casino keno game uses a circular glass enclosure called a "bubble" containing 80 ping pong-like balls which determine the balldraw result. Each ball is imprinted with a number 1 through 80. During the balldraw, a blower pushes air into the bubble and mixes the balls. A "caller" presses a lever opening a tube, where the balls lift one at a time into a "V" shaped tube called the "rabbit ears". The caller and a "verifier" record each of 20 balls drawn, and the computerized keno system calculates all wagers based on the numbers drawn.

Players wager by marking an "X" over the "spot" choices on a blank keno ticket form with 80 numbered selection boxes (1 to 80). After all players successfully place their wagers, the casino draws 20 balls (numbers) at random. Some casinos automatically call the balldraw at preset timed intervals regardless of whether or not players are waiting to place a wager.

Each casino sets its own series of pay scale choices called "paytables". The player is paid based on how many numbers drawn match the numbers selected on the ticket and according to the paytable selected with regard to the wager amount.. Players will find a wide variation of keno paytables from casino to casino and a large deviation in the house edge set for each of those paytables. Additionally, each casino typically offers many different paytables and specialty keno bets for customers to choose from, each with its own unique house edge. No two casinos' keno paytables are identical. There are several Reno and Las Vegas casinos offering as many as 20 or 30 different paytables from which the player can choose.

The house edge ranges from less than 4% to 66%. The typical house edge for non-slot casino games is between 0% and 5%

## Keno history

Keno originated in ancient China in the Han Dynasty between 205 and 187 B.C. Chinese immigrants brought the game to America in the 19th century.

The word "keno" has French or Latin roots (Fr. quine five winning numbers, L. quini five each), but by all accounts the game originated in China over 2000 years ago. Legend has it that the invention of the game saved an ancient city in time of war, and its widespread popularity helped raise funds to build the Great Wall. According to one source, results of keno games in great cities were sent to outlying villages and hamlets by carrier pigeon. Eventually, Chinese immigrants introduced keno to the West when they sailed across the Pacific to help build the American trans-continental railroad in the 19th century.

## Keno odds

Keno payouts are based on how many numbers the player choses and how many numbers are "hit", multiplied by the proportion of the player's original wager to the “base rate” of the 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 from casino to casino. Most casinos allow paytable wagers of between 1 and 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 3,535,316,142,212,180,000 to be exact). If every person now alive played one keno game every single second of their lives, there would be about one solid 20 jackpot-winning ticket to date. If all these possible keno tickets were laid end to end, they would span the Milky Way galaxy -- and only one of them would be a winner. Even though it is virtually impossible to hit all 20 numbers on a 20 spot ticket, the same player would typically also get paid for hitting “catches” 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:

20 Spots 0 Hits: 1 In 843.380 Or 0.11857057%
20 Spots 1 Hits: 1 In 86.446 Or 1.15678605%
20 Spots 2 Hits: 1 In 20.115 Or 4.97142576%
20 Spots 3 Hits: 1 In 8.009 Or 12.48637168%
20 Spots 4 Hits: 1 In 4.877 Or 20.50318987%
20 Spots 5 Hits: 1 In 4.287 Or 23.32807380%
20 Spots 6 Hits: 1 In 5.258 Or 19.01745147%
20 Spots 7 Hits: 1 In 8.826 Or 11.32954556%
20 Spots 8 Hits: 1 In 20.055 Or 4.98618021%
20 Spots 9 Hits: 1 In 61.420 Or 1.62814048%
20 Spots 10 Hits: 1 In 253.801 Or 0.39401000%
20 Spots 11 Hits: 1 In 1423.822 Or 0.07023351%
20 Spots 12 Hits: 1 In 10968.701 Or 0.00911685%
20 Spots 13 Hits: 1 In 118084.920 Or 0.00084685%
20 Spots 14 Hits: 1 In 1821881.628 Or 0.00005489%
20 Spots 15 Hits: 1 In 41751453.986 Or 0.00000240%
20 Spots 16 Hits: 1 In 1496372110.872 Or 0.00000007%
20 Spots 17 Hits: 1 In 90624035964.712 Or 0.00000000%
20 Spots 18 Hits: 1 In 10512388171906.553 Or 0.00000000%
20 Spots 19 Hits: 1 In 2946096785176811.500 Or 0.00000000%
20 Spots 20 Hits: 1 In 3535316142212173800.000 Or 0.00000000%

## Modern keno

The balldraw occurs at the "keno booth". The balldraw is typically determined by one of four devices:

2. "AKV": Automated blower in which the balls are encoded, read by a computer, then sent to a computerized keno system for processing
3. Random Number Generator: An electronic number selection device approved by gaming authorities
4. “Hand Cage”: A spinning the metal ball cage which rolls the numbers into a slot where they are validated by a person

"Keno runners" walk around calling, "keno!" and offer to carry players' wagers to the keno booth for processing. The keno runner is handed the wager payment and the “inside ticket” (keno wager forms filled out by the customer) and takes the wager and ticket to the keno counter for processing. The keno runner returns with an "outside" ticket, which is the official wager receipt. It is incumbent on the player to check the ticket for errors before the game balldraw is drawn. Any errors not corrected before the balldraw begins are not normally rectified and the "outside" ticket receipt governs in any disputes.

In modern keno, players are offered the option of playing multi-race keno, which books a keno ticket for a number of sequential keno races up to 1000. The races must always start on the next sequential race to be drawn. When the sequence of wagered games is finished, the player is able to redeem any winnings within the time constraints specified in the casino's rules booklet.

After picking wager numbers, recording them at the keno booth and obtaining the “keno ticket” (official wager receipt), the player watches the balldraw in progress as the spot (number) selections light either on an electronic keno board or on a video monitor. Keno displays are typically found throughout the casino and sometimes even appear on a television channel in casino hotel rooms. Nowadays, after playing keno at a participating casino, keno players can even take their multi-race keno tickets out of the casino to watch the live balldraw or check historical draw results at [www.kenousa.com] anywhere there is a computer and an Internet connection.

In the past, a winning ticket needed to be taken to the keno booth for collection immediately after the race results were posted, and drawings usually took place approximately every five or six minutes. In days of old, if the player tried to redeem a winning ticket after the next sequential race began, the ticket was declared expired and no money was paid out even if it won. Nowadays most casinos set their "late pay" window to accommodate the player, however there is great variation in the published late pay window from casino to casino. Tickets played for 21 races or more typically offer one year for collection in most major gaming jurisdictions. Tickets played for under 21 races have a great disparity of late pay rules from casino to casino. Keno players are wise to read the rules published in the host casino's keno paybook to determine when a keno pay will expire and become uncollectible. Gaming authorities require that all pay scales and keno rules be posted in a prominent location in keno areas.

An embellishment of keno is “way” keno or “combination” keno. When playing a way or combination keno ticket, the player circles groups of numbers within the spots marked and specifies combinations of groups which combine together to make different "ways". For example, if a player marks four numbers, and circles two groups of two spots each, a combination ticket could be made in which the gambler plays one 4 spot and two 2 spots (2-2). If an additional group of two were added and circled, the player could play ways 1/6, 3/4 and 3/2 (2-2-2), which at \$1.00 per way would create a \$7.00 per race wager. Serious keno players use great imagination on keno tickets to make complex combinations of groups and ways with varying numbers of spots in each group. There is literally no limit to the way players can circle spots of like or a differing number of spots and in the resulting “ways” they can choose to play on a keno ticket.

As alternatives to traditional paytables which offer the selection of 1 to 20 spots, a number of special paytables are available and are often offered as a wager choice. For example, with the Top/Bottom paytable the keno player does not select any spots. Rather, the player is betting that the balldraw top 40 and bottom 40 ball distribution will be uneven. Top/Bottom typically begins paying on a 7/13 or 13/7 ball distribution between the top half (1 to 40) and bottom half (41 to 80) of the keno grid and the payouts increase with each higher balldraw distribution disparity. The same principle is applied for the Left/Right or Odd/Even paytables. Other specialty paytables are Top Only, Bottom Only, Left Only, Right Only, Odd Only, Even Only, Parlay, Exacta, Edge, Square, or eXtra Million, which is proprietary to XpertX Keno Systems. However the traditional 1 though 20 spot pick is by far the most popular variety of live keno.

Lottery versions of keno are now used in many National Lotteries or state licensed Lotteries around the world. The games have different formulas depending on the wanted price structure and whether the game is slow (daily or weekly), or if it is a fast game with just minutes between the draws. The drawn numbers are typically published on TV for the slow games and on monitors at the point of sale for the fast games. A video keno machine sometimes has a greater customer edge than a traditional keno game. However, because live keno payouts are configurable at will by the host casino, some live keno paytables house hold percentages match or are even lower those for video keno machines, which almost always have fixed paytables that don't change.

## Detailed mathematical analysis

The version of Keno played in Maryland serves as a case study in the precise calculation of win probabilities and expected return—the latter referring to the result to be realized in the long run from each unit invested.

In Maryland, anyone may play keno at any of thousands of establishments that are wired with a television screen and an impossible-to-overlook, hot pink machine resembling a cash register. The player uses a pencil to complete a small slip; the attendant feeds the slip to the machine, which generates a computer-printed ticket that is protected from tampering via cryptographic checksum. Games—which are played every four minutes or so—can be viewed over the accompanying television monitor. The computer selects twenty numbers between one and eighty. The payout is calculated based upon how many numbers were chosen and how many were matched. Intriguingly, for the nine-spot and ten-spot games, there is a payout if the player fails to match any numbers—it obviously being an unusual event for zero of nine or ten selected numbers to match any of the twenty "dealt," so to speak, from the pool of eighty.

The probability that k of the n numbers chosen by the player, i.e.,

$P_n \left(k\right)$

occur in the twenty numbers chosen by the computer can straightforwardly be derived:

1. The number of possible outcomes equals the number of combinations of eighty numbers taken twenty at a time.
2. The number of ways in which k of the n numbers selected by the player occur in the twenty numbers selected (putatively at random) by the central Keno computer is equal to the number of ways in which k numbers can be chosen from a set of n numbers.
3. The number of ways in which the remainder of the numbers do not occur in the twenty numbers selected is given by the number of ways in which 20-k numbers can be chosen from a set of 80-n numbers.

Combining the foregoing, one finds that:

$P_n \left(k\right) = frac\left\{\left\{n choose k\right\} \left\{80-n choose 20-k\right\}\right\}$

The payouts for each result can be read from the Maryland keno Web site. For the purposes of our discussion, if the player participates in the n-spot game and ends up matching k of the twenty numbers selected, we will refer to that payout as:

$W_n \left(k\right).$

The expected payout for the n-spot game can be determined by summing, over all values of i from one to n (from zero to n if the game pays out in the case of zero numbers matched), the product of the payout for that result and the probability of occurrence of that result:

$sumlimits_\left\{i=0 or 1\right\}^\left\{n\right\}P_n \left(i\right) W_n \left(i\right)$

which could alternatively be represented as the inner product ("dot product") of the vector of probabilities and the vector of payouts.

One finds that the best game for the player is the three-spot game, which realizes an expected return of approximately 62 cents for every dollar invested, or approximately a 38% loss. The seven-spot game ranks close behind, returning just over 60 cents per dollar. Perhaps not surprisingly, despite the astonishingly high payoff for strong performance, the ten-spot game is by far the poorest from the player's perspective.

## References

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