# Gaming the Law of Large Numbers

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1 Gaming the Law of Large Numbers Thomas Hoffman and Bart Snapp July 3, 2012 Many of us view mathematics as a rich and wonderfully elaborate game. In turn, games can be used to illustrate mathematical ideas. In this article we discuss an adaptation of the game Liar s Dice that we will call Fibber s Dice. Not only is Fibber s Dice a fast-paced game which rewards gutsy moves and favors the underdog, it also brings to life concepts arising in the study of probability. In particular, Fibber s dice nicely shows the connection between counting, probabilities, relative frequency, and the law of large numbers. The Rules of Fibber s Dice We developed the rules for Fibber s Dice by directly adapting the rules of Liar s Dice found at [3]. These modifications make the game more suitable for the classroom. Setup and object of the game Each player starts with a cup and six dice. The winner is the first player to lose all of their dice through bidding, bluffing, and luck. Game play Each round starts with the players rolling their dice while using the cup to conceal the values. After each player examines their own dice, the first player makes a bid. This bid should represent an estimate of the total number of dice showing a particular value under all the cups on the table. However, there are two catches: All ones are wild and are counted as the value of whichever bid is made. You cannot bid on the number of ones showing. Thus a call of 7 fours is based on a prediction that there will be a total of at least 7 dice with a value of either four or one. Bids continue to the left increasing each time. There are two ways to increase the bid: One can increase the number of dice with any face value, two through six. One can keep the same number of dice and increase the face value. 1

4 Assessment Since we use this project in classes with varied mathematical backgrounds, we have tried several different assessment methods: Writing assignments where the students describe their strategies, including cases for both initial bids and trailing bids. Given a description of a specific game situation, the students should be able to discuss the merits of a particular bid. They should also be able to give an argument for either challenging a certain bid, or if not, what the following bid should be. Often we provide a table similar to Table 1 to aid in their reasoning. While the level of the students determines the computational complexity of the problems, most of the assessments we use are not entirely based on numerical answers. We look for student s understanding when using probabilities and expected values, which are measured by requiring the students to properly justify their answers through a description of their thought process. Connections to probability When introduced to probability, you are often presented with three notions: probability, relative frequency, and the law of large numbers. Fibber s Dice illustrates the connection between these ideas. The notions of probability and relative frequency are glued together via the law of large numbers. This states that the relative frequency of an event approaches the probability of the event as the number of outcomes observed approaches infinity. The idea is that if a coin is flipped a few times, you may not experience the same number of heads as tails but if you keep flipping the coin long enough, the ratio of heads (or tails) to the total number of flips will approach 1 2. While many games of chance such as poker and roulette can be analyzed using probabilities, the number of events occurring in a given game is too few for one to really make use of these computations. An unlikely hand in poker can still appear, and the seemingly unlikely event of 8 reds in a row in roulette can still happen, but you may play for a long time without witnessing this event. Roulette can easily be modeled in the classroom using a deck of cards (in place of the ball and wheel) and poker chips for betting. Unfortunately, the number of rounds you can get in an hour is small and the only lesson the students are sure to walk away with is that gambling only pays off for the casino. The probability of 8 reds in a row in roulette is approximately 1/400. Playing 10 rounds an hour, students would need to play for approximately 40 hours to see this event. What makes Fibber s Dice different from most games involving probability is that the effect of the law of large numbers can readily be seen while playing the game. When five people are playing Fibber s Dice, the game starts with 30 dice on the table. This game will include at least 165 dice rolls, and often games 4

6 the following, we look into a couple of specific situations comparing our three strategies. Similar investigations could be completed by the students. While value bidding is better than blind bidding, and advanced bidding seems like it should be the best, how can we be sure? Using Mathematica, we simulated 5000 rounds for each of these three bidding strategies. Each round was for five players with six dice each. We call a bid a winning bid if it would win if challenged. While the die value affects the strategy for bidding, the probability of a bid winning can be modeled using just the number of dice. We gathered data by looking at the winning and losing bids in these rounds. Our data is summarized in the following graph. 1.0 Relative Frequency Bid In the graph above, the solid line is a continuous representation of the relative frequency of making a winning bid using blind bidding. The dashed line represents the relative frequency of making a winning bid using value bidding. Notice that when there are thirty dice on the table, value bidding allows the player to bid 1 die higher than blind bidding while maintaining a comfortable relative frequency of.6 of winning a challenge. It is somewhat of a surprising fact that from this perspective, advanced bidding gives no significant advantage over value bidding. To see how advanced bidding gives an advantage we look at the game from a new viewpoint. Favors the Underdog As shown in the situations described above, the advantage in Fibber s Dice goes to the players with the most dice, since these players have the most information available to them. However, these players are furthest from winning! As a player loses dice (which is good!), value bidding becomes less effective. This makes Fibber s Dice s more fun to play as it gives the underdog a chance to catch up. We witness this advantage to the underdog through computer experiments. Suppose that there are five people playing, four of which are doing quite well, 6

7 say they have only three dice each. Suppose further that the fifth person is doing considerably less well, and still has all six of their starting dice. Let s suppose that someone with three dice is bidding. What are their prospects? Consider the following bar chart: Blind Bidding n/3 Value Bidding n/3 looking Advanced Bidding n hand + hand 3 The teal bar represents the player s probability of making a winning bid, via the three strategies, blind bidding, value bidding, and advanced bidding. The gray bar represents the probability of the next player making a higher winning bid assuming that the first player was correct. In this case we assume that both players have three dice. From our data we see that value bidding gives the greatest probability of stating a winning bid. However, if you are using value bidding and if you are correct, then the next player also has a high probability of being correct with their next bid. Value bidding does not put much pressure on the next player. If the next player does not challenge you cannot lose dice, meaning you won t win! Now suppose that the underdog, the person with six dice is the first player, 7

8 and someone with 3 dice is following them: Blind Bidding Value Bidding n/3 n/3 looking Advanced Bidding n hand + hand 3 Using blind bidding leaves the probability unchanged. We see an improvement using value bidding. When the underdog uses advanced bidding the second players probability of stating a winning bid falls dramatically. In both of the above graphs, it is difficult to see the difference between advanced bidding and blind bidding. Recall, the extra information used to make an advanced bid allows the player to make a higher bid with the same probability as the other bidding techniques. This is how the second player s probability of a winning bid is being forced down. This higher bid will cause the second player to either challenge the first player because the bid is above n/3, or make a high bid with which they are uncomfortable. Challenging Strategies The question now becomes, When should I challenge? A basic strategy would be to challenge whenever a bid is made that is greater than n/3. How much higher should you bid? If there are n dice on the table, the probability that exactly a quantity of q of any value are showing, including wilds is: ( ) q ( ) n q 1 2 P (q) = ( n C q ) 3 3 The second factor gives us the probability of q dice rolling the chosen value or a wild. The third factor gives the probability of n q dice not rolling the chosen value or a wild. To see the role of the first factor, consider the following 8

9 diagram: } {{ } n dice We know that q of the dice in this picture are showing the chosen value or wilds. The factor n C q gives us all the ways to choose q of the n positions for these dice. In Fibber s Dice, we are more interested in the probability that q or more dice show a given value. This probability is given by: P sum (q) = n ( n C x ) x=q ( 1 3 ) x ( ) n x 2 3 While playing Fibber s Dice, we are looking for a comfort zone. A comfort zone is a range in the probabilities which seem good, not too risky and not too safe. Remember that we lose dice by being challenged and winning. As the round progresses, the bidding continues increasing, moving farther from the value n/3. Each player may have their own comfort zone, but eventually the bids get too high, or the probabilities get too low, and the player whose turn it is will prefer to challenge than make a bid of their own. Unfortunately, P sum (q) is not a calculation most people can do while playing. Of course, students could easily compute the values using a spreadsheet. We used Maple 11, [2], to construct the approximate probabilities. The following table is a portion of what we give to students so they can find their comfort zone as a bid quantity. (n) number n/3 P sum (q) of dice Table 1: Some q values for specific n and P sum (q) For example, when 54 dice are on the table, there is a.4 probability that some face value or 1 appears on 19 or more dice. For a more conservative player, this might be the upper limit of their comfort zone. A more adventurous player might be comfortable with the.25 probability associated with 20 or more dice. 9

10 Conclusion It is our hope that you try out Fibber s Dice, have fun, be daring, and experience the law of large numbers. In our classes, we like to ask the question, Does knowing the mathematical theory behind the game make the game more or less fun? We know our answer. References [1] J.S. Milton and J.C. Arnold, Introduction to probability and statistics: Principles and applications for engineering and the computing sciences, 3rd ed., Mcgraw Hill, [2] Waterloo Maplesoft Ontario, Maple 11, [3] Wikipedia, Liar s dice wikipedia, the free encyclopedia, 2008, [Online; accessed 8-August-2008]. [4], Perudo wikipedia, the free encyclopedia, 2008, [Online; accessed 8-August-2008]. 10

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