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William Chon 22600008Stat 157 Project, Fall 2014Near Misses in BingoI.IntroductionA near miss is a special kind of failure to reach a goal, in which one is close to beingsuccessful. Near misses can arise in a variety of scenarios that involve components ofboth skill and luck. In games of pure chance, for example, a person that purchases alottery ticket with the following combination (5 14 15 18 33) will experience a nearmiss if the actual lottery drawing turns out to be (5 14 15 18 34). Likewise, we cancommonly observe near misses in games of skill, such as bowling. A near miss can begetting eleven strikes in a row, then hitting 9 pins on the last roll for a total score of299 rather than a perfect 300.In the case of skilled-based scenarios where we have a near miss, it can be a prettyrobust indicator of future success. However, for games of pure chance, a near missprovides no information that can increase the likelihood of future success. Researchalso shows that in both skill-based and chance-based scenarios, players frequentlyinterpret an occurrence of a near miss as an encouraging sign, confirming the playerβsstrategy and raising his or her hopes for future success.Because near-misses are ubiquitous across many different skill-based and luck-basedgames, there are many avenues that I can explore. For this project, I want to firstexplore the psychology behind near misses. I would then like to explore the frequencyof near misses in bingo, a popular game of chance, and examine how the psychologyof near misses may potentially affect a beginning bingo player.

Chon 2II.Psychology of Near MissesNear misses are a very interesting topic because they arise in everyday activities. Inscenarios involving transportation safety and damage prevention, near misses can betaken as valuable zero-cost learning opportunities. There are many systems in placethroughout the world that have improved safety through the anonymous reporting ofnear-miss incidences. A few examples include1: In 2005, the National Fire Fighter Near-Miss Reporting System wasestablished, funded by grants from the U.S. Fire Administration and FiremanβsFund Insurance Company, and endorsed by the International Associations ofFire Chiefs and Fire Fighters. Any member of the fire service community isencouraged to submit a report when he/she is involved in, witnesses, or is toldof a near-miss event. The report may be anonymous, and is not forwarded toany regulatory agency AORN, a US-based professional organization of perioperative registerednurses, has put in effect a voluntary near miss reporting system (coveringmedication or transfusion reactions, communication or consent issues, wrongpatient or procedures, communication breakdown or technology malfunctions.An analysis of incidents allows safety alerts to be issued to AORN members. CIRAS (the Confidential Incident Reporting and Analysis System) is aconfidential reporting system modelled upon ASRS and originally developedby the University of Strathclyde for use in the Scottish rail industry. In the United Kingdom, an aviation near miss report is known as an "airprox",an air proximity hazard, by the Civil Aviation Authority. Since reportingbegan, aircraft near misses continue to declineBeyond near-miss reporting, studies have shown that commercial gambling systems,particularly instant lotteries and slot machines, are contrived to ensure a higherfrequency of near misses than would be expected by chance alone. There are two main1http://en.wikipedia.org/wiki/Near miss %28safety%29

Chon 3reasons why near misses are significant in games of chance: they add excitement andencourage future play.In an experiment conducted at Exeter, the researchers simulated five horses racingrepresented by five dots moving across a screen form a start line to a finish line. Thetest subjects found the close races, where the dots moved at the same fixed paceexcept for small increments of random forward movements, to be the most interestingand exciting. On the other hand, the test subjects found the decided races, where ahorse separates early on from the pack, to be the least interesting and the worst.In another famous experiment, researchers wanted to study how test subjects wouldrespond to near misses in slot machines. In this experiment, the slot machines thatwere used had three wheels: first wheel had 70% red and 30% green logos, secondwheel had 50% red and 50% green logos, and third wheel had 30% red and 70% greenlogos. The winning combinations were either assigned to be three red logos or threegreen logos, and the probability of winning in each case is 10.5%. In addition, becausethe wheels stop from left to right, the winning combination of three reds would havea much higher frequency of near misses than the winning combination of threegreens. Thus, by prescribing the test subjects to the winning combination of threereds, we are inducing them to play on the slot machine with a higher incidence ofnear-misses.Using these slot machines, and experiment was carried out on forty-four male highschool students, in which they were each given 100 nickels to play with, and received40 cents for each winning combination. The players could choose to stop at any time,and retain half of their earnings. The results showed that the test subjects who wereplaced in the near-miss group played significantly longer on average, suggesting thatnear-misses encourage future play.

Chon 4III. Introduction to BingoA. Basic Rules of BingoBingo is a game of chance played with randomly drawn numbers which players matchagainst numbers that have been pre-printed on 5x5 matrices. The matrices may beprinted on paper, card stock or electronically represented and are referred to as cards.Many versions conclude the game when the first person achieves a specified patternfrom the drawn numbers. The winner is usually required to call out the word "Bingo!"which alerts the other players and caller of a possible win. All wins are checked foraccuracy before the win is officially confirmed at which time the prize is secured anda new game is begun.2A typical Bingo game utilizes the numbers 1 through 75. The five columns of the cardare labeled 'B', 'I', 'N', 'G', and 'O' from left to right. The center space is usually markedwith a "Free Space", and is considered automatically filled. The range of printednumbers that can appear on the card is normally restricted by column, with the 'B'column only containing numbers between 1 and 15 inclusive, the 'I' columncontaining only 16 through 30, 'N' containing 31 through 45, 'G' containing 46 through60, and 'O' containing 61 through 75.Figure 1: Example Bingo Card2http://en.wikipedia.org/wiki/Bingo (U.S.)

Chon 5Examining the card further, we can calculate the number of different bingo cardpermutations we can form using the rules stated above:# ππ πππππ’π "B" πΆπππ’πππ 15 14 13 12 11 360,360# ππ πππππ’π "I" πΆπππ’πππ 15 14 13 12 11 360,360# ππ πππππ’π "N" πΆπππ’πππ 15 14 13 12 32,760# ππ πππππ’π "G" πΆπππ’πππ 15 14 13 12 11 360,360# ππ πππππ’π "O" πΆπππ’πππ 15 14 13 12 11 360,360πππ‘ππ # ππ π’ππππ’π π΅πΌππΊπ πππππ πππππ’ππ‘ ππ πππ πππ£π ππππ’πππ 360,3604 32,760 552,446,474,061,128,648,601,600,000 πππ ππππA player wins by completing a row, column, or diagonal. There are a total of 12configurations that will complete a BINGO.There are also additional winning possibilities on top of the 12 winning configurationsstated above. Here are a few common modified versions of bingo: Postage Stamp: 2x2 square of marked squares in the upper-right-hand corner Corners: Another common special game requires players to cover the fourcorners Roving 'L': requires players to cover all B's and top or bottom row or all O's andtop or bottom row Blackout: cover all 24 numbers and the free space

Chon 6B. Simple Bingo StatisticsFor this project, I would like to concentrate on bingo with the original winningconditions. In other words, a player wins only when they complete a row, column ordiagonal.To get a better understanding of the expected number of turns it takes before a singleplayer achieves a BINGO, we observe the following distribution of turns it takes toachieve a BINGO with one player with ten thousand 289.86Figure 2: Ten thousand simulations of BINGO with one player. Average number of turns to achieve BINGO 42.3 (red line).Standard deviation: 9.86.

Chon 7From the results of my simulation, we see that a bingo game is slightly skewed to theright (longer games are more common). Overall, we should expect 42 to 43 numbersto be called before a single player achieves a BINGO.C. Bingo Hall SizesBingo starts to get a lot more interesting if we decide to add more players to the game.With more players, the prize pools increase in size, and the largest bingo halls inAmerica can oftentimes boast a jackpot size in the neighborhood of one-hundredthousand dollars.After sorting through Yelp and various local bingo review websites for the capacitiesof bingo halls across the United States, I have narrowed bingo games down to threemain sizes:1. Mega Bingo Halls & Casinos: 1,200 playersa. Foxwoods Resort Casino (4,000 seats), San Manuel Indian Bingo & Casino(2,500 seats), Cherokee Tribal Bingo (2,000 seats), Penobscot High StakesBingo (1,800 seats), Ft. McDowell Casino and Radisson Hotel (1,700 seats),Potawatomi Bingo Casino (1,354 seats)2. Large Bingo Halls & Casinos: 300 playersa. Smaller Casinosb. Larger local bingo halls3. Local Bingo Halls & Events: 100 playersa. Smaller local bingo hallsb. Retirement home gamesc. Community center games

Chon 8IV. Near Misses in BingoAfter extensive searches online, I found little to no information regarding near missesin the widely played game of bingo. I thought it would be an interesting task toexplore near misses in bingo games of various sizes. In particular, a near miss is whena bingo card is one square away from a BINGO when another player in the samegame wins. In this case, I want to explore a few questions:1. How many numbers should we expect to draw before a BINGO is reached,comparing across various game sizes?2. With what frequency does a near miss occur in a bingo game, and how does itcompare across games of different sizes?3. What is the relationship between how many numbers are drawn before aBINGO and the number of near misses there are for a particular game?Using my initial research on bingo game sizes as a range, I plan to explore thefrequency of near misses for games between 100 to 1,200 players in 100 playerintervals (i.e. 100, 200, 300, , 1,200).In order to answer the questions above, I found that simulations would be the mostpractical approach in finding an answer.3 The results of my simulations, and theresponses to my questions above are answered in the following sections.D. Expected Number of Draws Before a BINGOThe expected number of draws before a BINGO is reached decreases as the numberof players playing the same bingo game increases. In addition to that, the variance ofdraws required before a BINGO is reached decreases as the number of playersincreases.3Specifications and code used for the simulations can be referenced in the appendix. Each game specification wasrun through 1000 simulations for a total of 12,000 simulations and 7.8 million unique bingo cards.

Chon 9Game d Draws Until 00511.95011.99511.58511.610SD for Draws Until 2.7452.6162.471Table 1: Expected number of draws until a bingo is reached, with varying game sizesLooking for closely at the games with 100, 300, and 1,200 players, we observe thefollowing distribution of draws required to reach a BINGO:Figure 3: Distribution of draws before a BINGO was reached. Light blue 100 player games, green 300 player games, yellow 1,200 player games

Chon 10Comparing the histograms above, we can see that the number of draws until aBINGO is reach is distributed roughly symmetrical around the average.Plotting this data in a scatterplot, we can see that the relationship between theexpected draws until a BINGO is reached and the number of players in the game canbe represented by the following power function:# π·πππ€π πππ‘ππ π΅ππππ 34.483 (# ππππ¦πππ ) 0.155( 1)Draws Until BINGO18.00017.000Numbers Drawn16.00015.00014.00013.000y 34.483x-0.155RΒ² ame Size (Players)Figure 4: Expected number of draws until a bingo, across various game sizes. Each data point is the average of 1,000 simulationsE. Number and Frequency of Near Misses in Bingo GamesThe expected number of near misses increases as a linear function as the number ofplayers playing the same bingo game increases (figure 5). In addition to that, thevariance of the number of near misses increases as the number of players increases.

Chon 11Game d Number of 025.83529.39034.34031.11035.015SD for Number of Near 3418.95227.02724.50224.885Table 2: Expected number of near misses per game, with varying game sizesExpected Number of Near Misses, By Game SizeExpected Number of Near Misses3833y 0.0221x 9.6966RΒ² 0.98062823181380200400600800100012001400Game Size (Number of Players)Figure 5: Expected number of near misses in a game of bingo, across various game sizes. Each data point is the average of 1,000simulationsThere is, however, a problem with comparing the number of near misses acrossgames of different sizes. If games are larger, then we should expect to see more nearmisses, just due to the fact that there are more players playing the game. In orderto correct for increasing game sizes, we simply divide the expected number of nearmisses by the number of players playing in each game to determine the expected

Chon 12frequency or probability of a near miss occurring across bingo games of differentsizes. Now, we observe the following:Game d Near Miss .0330.0340.0280.029SD for Near Miss .0210.0270.0220.021Table 3: Expected probability of near misses per game, with varying game sizesFigure 6: Distribution of frequency of near misses per bingo game. Light blue 100 player games, green 300 player games,yellow 1,200 player games

Chon 13Probability of a Near Miss, By Game Size11.00%Near Miss Frequnecy10.00%9.00%8.00%7.00%6.00%5.00%4.00%y 0.8941x-0.485RΒ² 0.99673.00%2.00%0200400600800100012001400Game Size (Number of Players)Figure 7: Expected probability of a near miss in a game of bingo, across various game sizes. Each data point is the average of1,000 simulationsInterestingly, we observe that the expected probability of a near miss decreases asthe number of players playing the same bingo game increases. In addition to that, thevariance of the probability of a near miss decreases as the number of playersincreases. Once again, we are able to model the relationship between the probabilityof a near miss and game size:ππππππππππ‘π¦ ππ ππππ πππ π 0.8941 (# ππππ¦πππ ) 0.485( 2)F. Probability of a Near Miss, Given Number of Draws for BINGOLastly, I want to explore the probability of a near miss, given the number of drawsit took to win a game of bingo, regardless of the size of the game. This will help usget a better understanding as to why smaller games of bingo have a higherprobability of obtaining near misses.

Chon 14Probability of a Near Miss, Given the Number of Truns UntilBINGOExpected Probability of a Near Miss60%y 0.0033e0.1782xRΒ² 0.965750%40%30%20%10%0%49141924Number of Turns to Achieve BINGOFigure 8: Expected probability of a near miss in a game of bingo, as a function of the number of turns to achieve a BINGO,regardless of game sizes. The data points are an aggregate of 12,000 simulations.As we can see, in figure 8, the probability of getting a near miss increasesexponentially (3) as a function of the number of turns to achieve BINGO.ππππππππππ‘π¦ ππ ππππ πππ π 0.033 π 0.1782(# π·πππ€π πππ‘ππ π΅πΌππΊπ)( 3)Piecing the results of my simulation together, we saw earlier that the smaller thebingo game is, on average we should expect more draws until a bingo is reached.Looking at the results of the graph above, it reaffirms our earlier analysis on theexpected number of draws until a BINGO is reached as well as the expectedfrequency of BINGOs in a particular bingo game, using data across all game sizes.

Chon 15V.ConclusionAfter considerable research, we have significant evidence to suggest that humanstend to respond positively to near misses in games of both skill and chance, as nearmisses are interpreted as an encouraging sign. In games of chance, near misses arewidely believed to encourage future play, even though the probability of winningremains constant from trial to trial.After running simulations across bingo game sizes of 100 players up to 1,200 players,we were able to determine that the number of near misses in a bingo game increaseslinearly as a function of the number of players in the bingo game. This means that asthe size of the bingo game increases, the frequency of near misses decreases in amanner that can be represented by a power function.In the United States, we see that smaller games of bingo are much more popular.Whether this may be a result of convenience (itβs hard to find a lot of players as wellas venues for larger games), the fact is that players will win and get near misses moreoften in smaller games than larger games β which may add a greater factor ofexcitement than just a large jackpot number. If you want to play bingo for fun, thenI would highly suggest playing in smaller games as youβll have the thrill of gettingcloser to a win more often than in the larger games.VI. AppendixThe following two screenshots show the code that I used for the simulations. Here Ishow the simulations of 100 player games as an example, with the other simulationshaving the exact same structure.

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Near Misses in Bingo I. Introduction A near miss is a special kind of failure to reach a goal, in which one is close to being . such as bowling. A near miss can be . card stock or electronically represented and are referred t