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Using the On-line Dice-Based Golf Game GOLO to Illustrate Probability by John Gabrosek & Paul Stephenson Grand Valley State University Department of Statistics March 23, 2010 Rules of Golf A round of golf consists of 18 holes split into the front nine and the back nine. The object on any hole is to get the ball from the tee to the green and into the hole in as few strokes as possible. Golf holes are either par 3, par 4, or par 5. The par is the expected number of strokes that it will take for a very good golfer to hit the ball from the tee into the hole. Golf Terminology – Good Stuff Eagle – getting the ball into the hole in two less strokes than par In GOLO an eagle is denoted with a star. Golf Terminology – Good Stuff Birdie – getting the ball into the hole in one less stroke than par In GOLO a birdie is denoted with a circle. Golf Terminology – Good Stuff Par – getting the ball into the hole in the expected number of strokes In GOLO a par is denoted with a square. Golf Terminology – Bad Stuff Bogey - getting the ball into the hole in one more stroke than par. Double Bogey - getting the ball into the hole in two more strokes than par. Triple Bogey - getting the ball into the hole in three more strokes than par. ETC……….. Playing GOLO Each “golfer” is given nine 12-sided dice to roll. There are two par 3 dice, five par 4 dice, and two par 5 dice. The dice are placed into the cup, shaken up, and rolled onto a flat surface. Playing GOLO Each time the dice are rolled the golfer MUST remove at least 1 die. Any dice removed are then set aside and not used in subsequent rolls. A player continues to roll until he or she has used all nine dice and thus has completed nine holes. A player’s nine-hole score is the sum of the nine dice that have been set aside. GOLO Demonstration Playing the game of GOLO Classroom Application related to Strategies & Descriptive Statistics One potential activity that can be developed allows students to compare the following two strategies: Strategy 0 (No strategy): Each student rolls all nine GOLO dice and sums them to obtain a 9-hole score. Repeat this twice to obtain an 18-hole score. Strategy 2 (Lowest die): Each student rolls all nine GOLO dice and removes the best die relative to par. (If there are multiple dice with the same score relative to par, then any one, but only one, of the dice is removed.) The remaining dice are placed in the cup and rolled again. This procedure continues until all nine dice have been removed. The scores for two nines are summed to obtain an 18-hole score. Classroom Application Two example games with strategies 0 & 2: Strategy 0 (No-Strategy) Strategy 2 (Lowest Die) Game Front 9 Back 9 Score Front 9 Back 9 Score 1 50 57 107 32 33 65 2 46 44 90 40 35 75 Classroom Application Stem-and-leaf display for simulated results (60 trials with each strategy) Strategy 2 Strategy 0 6 124 6 5555667777789999 7 000000001111111122233334 7 5556666777788999 2 8 3 996 8 4444333322211100 9 999987766666555 9 444333332211000 10 98776655 10 00 11 Classroom Application This application can be used to introduce a variety of descriptive and inferential procedures for two independent samples including: side-by-side boxplots confidence intervals for independent samples hypothesis testing for independent samples The GOLO Dice There are two red par 3 dice. We call these die Par 3A and Par 3B. The twelve equally-likely faces on the dice are numbered as follows: Par 3A – 1, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 8 Par 3B – 2, 3, 3, 3, 4, 4, 5, 5, 6, 6, 6, 7 The eagle/birdie are denoted with a yellow box, and the pars are denoted with a green box. The GOLO Dice There are five identical white par 4 dice. The twelve equally-likely faces on the dice are numbered as follows: Par 4 – 3, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 8 The GOLO Dice There are two blue par 5 dice. We call these die Par 5A and Par 5B. The twelve equally-likely faces on the dice are numbered as follows: Par 5A – 3, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 10 Par 5B – 4, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9 Binomial distribution Consider the fact that each roll can be thought of as a series of trials where a success on any given die (or trial) can be defined as getting a par or better on the up-face. Suppose that N dice are rolled, the probability that X dice are par or better follows a binomial distribution. Geometric distribution Consider the fact that many games end by needing to roll exactly one die. Suppose that your competitor inexplicably leaves the table when you have one die left to roll and you are in need of at least a par to win. Let R denote the number of rolls it will take for you to roll a par or better. Then R follows a geometric distribution. Discussion regarding GOLO Any questions ???