Gambler s Ruin Lesson Plan

Transcription

1 Gambler s Ruin Lesson Plan Ron Bannon August 11, 05 1

2 Gambler s Ruin Lesson Plan 1 Printed August 11, 05 Prof. Kim s assigned lesson plan based on the Gambler s Ruin Problem. Preamble: This lesson plan is being developed for pre-college computer science students who are familiar with a procedural language. The material here will be presented using standard command line C++ on a Mac OS X platform, 2 but can easily be modified to run under other operating systems. I believe this material can be covered in five typical (50 minute session) class meetings, and will require at least one hour of homework per class session. Teacher s can visit gamblers-ruin.blogspot.com to get help, or to download this lesson plan and example ASCII code. I can also be contacted via at ron.bannon@gmail.com, and I welcome productive exchange of ideas, especially from students who actually worked through the material. Disclaimer: This lesson plan analyzes a gambling problem and playing an actual incidence of this game may be a violation of law. Please do not encourage your students to gamble, and emphasize that it is illegal to do so. Before using this lesson plan you may want to clear its content with your supervisor. 1 Session I: Playing a game-of-chance. The main goal is to simulate an actual game-of-chance to see what the outcome is. At first the simulation will involve your students, but then we ll speed-up the process by using a computer s random number generator. This example should help motivate students (and teachers) to use similar methods to explore, develop and implement probability models using a procedural language that has a robust random number generator. The problem is usually expressed in a very abstract manner as follows: The Gambler s Ruin 3 Two gamblers, A and B, bet on the outcomes of successive flips of a coin. On each flip, if the coin comes up heads, A collects 1 unit from B, whereas if it comes up tails, A pays 1 unit to B. They continue to do this until one of them runs out of money. If it is assumed that the successive flips of the coin are independent and each flip results in a head with probability p, what is the probability that A ends up with all the money if he starts with i units and B starts with N i units? However, I think it is best to avoid this description and instead take a specific case and divide the class into pairs in order to simulate an actually event. Give each pair of students twenty pennies and one dime. Then tell each pair to start playing the game as follows. 1 This document was prepared by Ron Bannon using L A TEX 2ε. Source and pdf are available by ing a request to ron.bannon@gmail.com. 2 BSD UNIX based OS. 3 From Sheldon Ross s textbook: A First Course in Probability, Seventh Edition, Prentice Hall, 02, ISBN: , pp

3 1. One member from each team will be asked to flip the dime, this person will be arbitrarily called A and the other B. 2. Give A fifteen pennies and B five pennies. Person A should have fifteen pennies and one dime, and person B should have five pennies. 3. Person A should flip the dime, if it is heads, A should take one penny from B, if it is tails, B should take a penny from A.. Repeat playing until one player has all twenty pennies, record which player wins. 5. Encourage students to repeat this game as many times as they like. This of course may be quite chaotic, and you may need to adapt this game for your particular class. It should also be emphasized while the students are playing that three possibilities exists: player A wins, player B wins or the game never stops. Actually, depending on time, many of your pairs may fail to finish a run. In any case, the game should prove fun, but still encourage your students to continue playing with others. Some students will complain that their particular dime is biased, or that their partner s know how to toss a dime to get a desired result, and you should not suggest otherwise. 2 Session II: Does random mean pure chance? It is hard to know if randomness is well understood, even by those of us who are supposed to know the difference. It is all about language and how it is used. Words get used imprecisely and their meaning is often not well understood just think of mathematicians saying: the rational numbers, although infinite, are countable, while the irrational numbers, also infinite, are uncountable, etc., ad nauseam. So often are words used to differentiate the subtlety between concepts that many of us are perplexed when common words are used in a very demanding and precise field such as mathematics. However, here you should emphasize that computers, although anything but random, have fairly good algorithms for generating random numbers. Present the following snippet of C++ code (Listing 1) as a fairly robust 5 way to generate a sample random number from the open interval (0, 1). Listing 1: Uniform Random Variable (0, 1) 1 double uniform ( void ) 2 { 3 long random_integer ; 5 do random_integer = rand (); 7 while ( ( random_integer == RAND_MAX ) ( random_integer == 0) ); 8 9 // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET // THE SET NOW HAS TWO LESS ELEMENTS return ( static_cast <double >( random_integer ) / ( RAND_MAX - 2)); 13 } // uniform Here it might be helpful to explain that rand() is a standard C++ function and the students will need to use cstdlib library in order to call rand(). The function rand() will return an integer, It must be emphasized that this is a game and no one is to walk away with the money! 5 See for more information about random numbers. 3

4 at random, from 0 to some machine specific maximum integer call RAND MAX. This double uniform(void) function s sole job is to return one random (equally likely) number from the open interval (0, 1). If all numbers are equally likely on the open interval (0, 1) it seems reasonable that half the numbers returned from double uniform(void) will be greater than 0.5 and the other half will be less than 0.5. This, of course, will allow us to simulate a simple fair coin toss, where heads will be equivalent to double uniform(void) function returning a number greater that 0.5, and tails will be equivalent to double uniform(void) function returning a number less than or equal to 0.5. As an assignment have your students write a simple C++ program that simulates a fair coin toss. Specify that they should generate a string or Hs and Ts that has length. Here is an example (Listing 2) of what a typical student should produce. Listing 2: Twenty flips of a fair coin // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET 1 // THE SET NOW HAS TWO LESS ELEMENTS 23 int i; // counter variable 2 25 for ( i = 1 ; i <= ; i++ ) 2 if ( uniform () > 0.5 ) 27 cout << "H "; 28 else 29 cout << "T "; 30 cout << "\n"; } // main function The output should be something like this: T T H T H T T H H H T H H T T H H T T T. 3 Session III: Simulating Class 1 s game on a computer. By now your students should have a full grasp of the game. Here it s imperative that the students know the game well enough to actually program one run, where we just want to know the outcome: A won, or B won. 7 As a reminder, here s how the game proceeds. 1. One member from each team will be asked to flip the dime, this person will be arbitrarily called A and the other B. Students who question the fairness because of the inclusion of 0.5 in the lower half should be encouraged to compute the length to see that they are actually the same. 7 In reality, this could be an infinite game where no one wins, but that s not going to happen on a machine because of the speed at which events happen, and the fact that this event has probability that is asymptotically zero.

5 2. Give A fifteen pennies and B five pennies. Person A should have fifteen pennies and one dime, and person B should have five pennies. 3. Person A should flip the dime, if it is heads, A should take one penny from B, if it is tails, B should take a penny from A.. Repeat playing until one player has all twenty pennies, record which player wins. 5. Encourage students to repeat this game as many times as they like. 8 As an assignment have your students write a simple C++ program that simulates one run of this game. Specify that they should output who wins the game, i.e. A or B. Here is an example (Listing 3) of what a typical student should produce. Listing 3: One run of the game // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET 1 // THE SET NOW HAS TWO LESS ELEMENTS 23 int A = 15; // number of pennies that A starts with 2 int B = 5; // number of pennies that B starts with 25 2 do 27 if ( uniform () > 0.5 ) 2 29 A ++; 30 B - -; 31 } 32 else 33 { 3 A - -; 35 B ++; 3 } 37 while ( A!= 0 && B!= 0); if ( B == 0 ) 0 cout << "A wins all!\n"; 1 else 2 cout << "B wins all!\n"; 3 } // main function Session IV: Increasing the number of games played. By now everyone should realize that A has an advantage and that B almost always loses. However, what if we were to repeat the game many times to see what A s real advantage is. As an assignment have your students modify their prior C++ program assignment (Listing 3) that simulates one run of this game, so that it runs 1,000,000 games and keeps tabs of how many wins both A and B receive. Emphasize that a physical simulation would take about 30 years of full time effort for two humans a colossal waste of time. However a fast personal computer will be able to simulate this fairly quickly. This, of course, is the bases for all probabilistic 8 It must be emphasized that this is a game and no one is to walk away with the money! 5

6 simulations... SPEED, SPEED, SPEED! However, an older machine may need several minutes to compute the results, so some students will need to be patient. Also, clearly specify that they should output the number of wins for both players. Here is an example (Listing ) of what a typical student should produce. Listing : One million runs of the game // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET 1 // THE SET NOW HAS TWO LESS ELEMENTS 23 int A = 15; // number of pennies that A starts with 2 int B = 5; // number of pennies that B starts with 25 int A_wins = 0; // number of wins for player A 2 int B_wins = 0; // number of wins for player B 27 int i; // counter for (i = 1 ; i <= ; i ++) 30 { 31 do 32 if ( uniform () > 0.5 ) 33 { 3 A ++; 35 B - -; 3 } 37 else 3 39 A - -; 0 B ++; 1 } 2 while ( A!= 0 && B!= 0); 3 if ( B == 0 ) 5 A_wins ++; else 7 B_wins ++; 8 A = 15; // number of pennies that A starts with 9 B = 5; // number of pennies that B starts with 50 } cout << "A wins " << A_wins << ".\n"; 53 cout << "B wins " << B_wins << ".\n"; 5 55 } // main function Outputs will vary slightly, my run produced: A wins 750,2; and B wins 29, Session V: Changing the parameters. This section will be divided into three parts, and your students should have a much better understanding of the overall problem and should be able to implement subtle changes in the code (Listing ) quickly. This is especially true if they re working independently and were capable of coding (Listing ) the last exercise. 5.1 Changing the beginning cash distribution. Again, the model (Listing ) will be easy to change, but first it might be nice to conjecture what might happen if we were to start out with different amounts. Looking at the last case we saw in

7 our simulation that the ratio of number of wins for player A to player B was roughly 3 to 1, and this was to be expected, mainly because it s the same ratio as the beginning cash distributions. So with this in mind ask the students to modify the starting cash amounts and re-run 9 their programs that simulate a million runs. Suppose for example that player A starts with 75 pennies and player B starts with 0 pennies. Before the students run the program they should be able to conjecture how many wins for each player first. The simulation should either confirm or deny this. Also have the students try starting player A with 90 pennies and player B with pennies and see what happens. Here is an example of what a typical student s simulation should produce. 1. Let A start with 75 pennies and B with 0 pennies. In a million runs of this game I d expect that B would win 0 1, 000, , 29 5 games, and A would win 75 1, 000, , games. When I change the program to reflect this change in starting values I get: A wins 28,537; and B wins 571,3. That s almost exactly what we expected to see. 2. Let A start with 90 pennies and B with pennies. In a million runs of this game I d expect that A would win 90 = 900, games, and B would win 1, 000, 000 = 0, games. When I change the program to reflect this change in starting values I get: A wins 899,950; and B wins 0,050. That s almost exactly what we expected to see. 5.2 Changing the fairness of the coin. The uniform distribution (Listing 1) that we presented earlier is the single most important distribution in probabilistic modeling, mainly because it is the basis for all simulated distibutions. It can easily be adapted to model complex distributions, but here we just need to create the occurrence of a biased coin. Let s say that we want the occurrence of heads to be twice a likely as an occurrence tail. Intuitively this means that heads will occur, on average, every two out of three; and tails will occur every one out of three. As an assignment have your students write a simple C++ program that simulates a biased coin toss, where we want the occurrence of heads to be twice a likely as an occurrence tail. Modifying (Listing 2) to reflect this biased coin s behavior, and generate a 1,000,000 samples, keeping tract of the number of heads and tails. Here s a example code (Listing 5) of what a typical student s work should look like. 9 Changing parameters can have a drastic impact on simulated run times, and your students should be reminded why by appealing to their understanding of the actual physical game. 7

8 Listing 5: 1,000,000 flips of a biased coin // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET 1 // THE SET NOW HAS TWO LESS ELEMENTS 23 int i; // counter variable 2 int H = 0; // number of heads 25 int T = 0; // number if tails for ( i = 1 ; i <= ; i++ ) 29 if ( uniform () > 1.0/3.0 ) // this will occur 2/3 rd of the time 30 H ++; 31 else 32 T ++; 33 cout << " The number is heads : " << H << "\n"; 3 cout << " The number is tails : " << T << "\n"; 35 3 } // main function The output should be something like this:,3 heads, and 333,337 tails, where, as expected we have, , 337 =, Biased coin and differing cash amounts. Again the model (Listing ) will be easy to change, but first it might be nice to conjecture what might happen if we were to start out with different amounts, and played with a biased coin. Suppose player A starts with 5 pennies, player B starts with pennies, and heads is favored 3 to 2. We already know that if the coin were fair player A would win one-third of the time, but now we re playing with a biased coin that favors player A. Have the students conjecture what s going to happen before running the simulation. I believe most students would venture a guess that A is actually favored to walk away with all twenty pennies, but I seriously doubt anyone would be able to give proper odds for this game. Here it should be emphasized that simulations can in fact be a very nice way to solve very difficult probability problems. Once again the game should be simulated a million times, where player A starts with 5 pennies, player B starts with pennies, and heads is favored 3 to 2. Here s an example (Listing ) of what a student s work should look like. Listing : 1,000,000 plays using a biased coin. 1 8

9 15 // THIS WILL EXCLUDE RAND_MAX AND 0 FROM THE SET 1 // THE SET NOW HAS TWO LESS ELEMENTS 23 int A = 5; // number of pennies that A starts with 2 int B = ; // number of pennies that B starts with 25 int A_wins = 0; // number of wins for player A 2 int B_wins = 0; // number of wins for player B 27 int i; // counter for (i = 1 ; i <= ; i ++) 30 { 31 do 32 if ( uniform () > 2.0/5.0 ) // heads is favored 3 to 2 33 { 3 A ++; 35 B - -; 3 } 37 else 3 39 A - -; 0 B ++; 1 } 2 while ( A!= 0 && B!= 0); 3 if ( B == 0 ) 5 A_wins ++; else 7 B_wins ++; 8 A = 5; // number of pennies that A starts with 9 B = ; // number of pennies that B starts with 50 } cout << "A wins " << A_wins << ".\n"; 53 cout << "B wins " << B_wins << ".\n"; 5 55 } // main function The outcome from this run indicates that player A wins 870,537 games and player B wins 129,3 games. So the odds favoring player A is almost seven to one. 11 Here it should be emphasized that games of chance that involve loss and gain are always stacked against the player! Take home message, DON T GAMBLE unless you re designing the game, but that s illegal. Only the State has a right to do this, and it s often referred to as a stupid tax 12 because it preys upon people who don t understand probability. Endnote Please refer interested and capable students to birthday-surprise.blogspot.com website to learn more about probabilistic modeling. This website will start with a rather simple and robust way to generate random samples (the basis of all probabilistic simulations) and allow for students to interact in a learning community that is not limited to the confines of a classroom. Again, I can be reached via at ron.bannon@gmail.com if questions or concerns remain about the material presented. Mathematically, we expect « «15 1, 000, , games, and once again our simulation is pretty damn close! 11 An 87% chance of winning any one game! 12 It can also be referred to as a poor tax because it preys upon people who are poor. For example, I have adult students who spend more on lottery than they do on food. 9