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SIMULATIONS

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Simulations are used by engineers, programmers, and other scientists to produce the probable results of an experiment or happening.

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COMING EVENTS SIMULATIONS IN GAMES. SIMULATIONS OF EVENTS OR FUTURE ACTIONS. SETTING UP SIMPLE SIMULATIONS ADVANCED SIMULATION – MONTE CARLO METHOD.

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FOCUS AND INQUIRY WHAT IS YOUR FAVORITE VIDEO OR COMPUTER GAME? WHAT DOES THIS “GAME” HAVE TO KNOW TO PLAY? WHAT STATISTICS ARE USED?

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MAJOR LEAGE BASEBALL SAMMY SOSA EDITION WHAT ARE THE STATISTICS FOR THE PITCHER: ERA, STRIKEOUT RATE… WHAT ARE THE STATISTICS FOR THE BATTER: BATTING AVERAGE, HOW BATTER DOES AGAINST CERTAIN PITCHER… IS THE BAT CORKED?

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GAME SIMULATION THE COMPUTER TAKES ALL OF THE INFORMATION (IN STATISTICAL FORM AND CALCULATES THE PROBABILITY OF AN EVENT HAPPENING. THE COMPUTER WILL CHOOSE WHAT WILL HAPPEN TO THE PLAYERS BY PROBABILITY.

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SIMPLE SIMULATION SITUATION: THE LAKERS ARE ONE POINT BEHIND. SHAQ IS FOULED WITH NO TIME LEFT ON THE CLOCK (TWO FREE THROWS) RUN 25 SIMULATIONS AND GIVE RESULTS

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POSSIBILITIES MAKES NO SHOTS—LOSES GAME MAKES ONE SHOT—TIES GAME AND INTO OVERTIME MAKES TWO SHOTS—WINS GAME

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STATISTICAL INFORMATION SHAQ IS A 63% FREE THROW SHOOTER NO OTHER STATISTIC IS NEEDED AT THIS TIME.

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SETTING UP A SIMULATION ON THE TI-83+

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USING THE PROB/SIM APPLICATION 1. CHOOSE RANDOM NUMBERS 2. DRAW TWO 3. RANGE: REPEAT YES 5. SET #’S 0-62 AS A POINT. (63 #’s) 6. SET #’S AS A MISS. (37 #’s)

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USING THE RANDOM NUMBER FUNCTION FIND THE RANDOM INTEGER FUNCTION: MATH-PRB #5 randInt (min#, max#, amount generated) randInt (0, 99, 2)—(1, 100, 2) will also work. SET UP PARAMETERS AS IN PROB/SIM. KEEP PRESSING ENTER 25 TIMES AND TALLY

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TALLY TIME AFTER YOU TALLY YOUR SIMULATIONS: HOW MANY WINS? HOW MANY TIES? HOW MANY LOSSES?

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WHY HAVE SIMULATIONS COST/DANGER NOT MATHEMATICALLY FEASIBLE NOT PHYSICALLY FEASIBLE

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EXAMPLES BOMBING OF IRAN (IRAQ EARLIER) DAMAGE DUE TO A POSSIBLE HURRICANE TO THE MIAMI AREA DAMAGE DUE TO A NUCLEAR EXPLOSION ON NEW YORK CITY FINDING THE POSSIBLE PROFIT WHEN A SALES CAMPAIGN IS STARTED

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GUIDED PRACTICE BUILD SIMULATIONS FOR THE FOLLOWING: RUN 25 SIMULATIONS FOR EACH: THE WEATHERMAN STATES THERE IS A 65% CHANCE OF RAIN NEXT FRIDAY—WILL IT RAIN FOR THE JULY 4 PARADE. THE SCHOOL POPULATION IS AS FOLLOWS: 43% WHITE; 37% HISPANIC; 15% BLACK; AND 5% OTHER. A COMMITTEE IS BEING FORMED – WHAT IS THE RACIAL COMPOSITION OF THE COMMITTEE—IF 12 MEMBERS ARE CHOSEN.

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AN ADVANCED SIMULATION MONTE CARLO SIMULATION

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FIND THE AREA OF THE WATER To further understand Monte Carlo simulation, let us examine a simple problem. Below is a rectangle for which we know the length [10 units] and height [4 units]. It is split into 2 sections which are identified using different colors. What is the area covered by the blue color?

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VIEW THE WAVES color? What Is The Area Covered By Blue?

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CONT. Due to the irregular way in which the rectangle is split, this problem is not easily solved using analytical methods. However, we can use Monte Carlo simulation to easily find an approximate answer. The procedure is as follows: 1. randomly select a location within the rectangle 2. if it is within the blue area, record this instance a hit 3. generate a new location and repeat 10,000 times

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CALCULATION BLUE AREA= # HITS x 40 UNITS 10,000 THIS CAN ALSO BE USED IN MS EXCEL USING CELLS AS POINTS OF CHOOSING BY THE COMPUTER. THERE ARE MANY DIFFERENT TYPES OF SOFTWARE THAT CAN CALCULATE THIS

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MONTE CARLO PRACTICE DESCRIBE HOW A MONTE CARLO SIMULATION WOULD WORK TO DISCOVER THE PERCENTAGE OF WATER ON THE EARTH’S SURFACE. USING 10,000 TRYS—HOW CAN YOU FIND THE RACIAL PERCENTAGE OF THE POPULATION OF NEW YORK CITY.

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HOW ABOUT 3-D THE SPREADSHEET, PAPER, AND IDEAS WITH TWO VARIABLES ARE TWO DIMINSIONAL. WHAT ABOUT A 3-D OBJECT? THREE VARIABLES? WHAT ABOUT VOLUME?

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PROBLEM HOW TO YOU KEEP AN APPLE FRESH ON THE SHELF OF A GROCERY STORE. IF IT SITS TOO LONG IT BECOMES SOFT AND MUSHY—NOT GOOD FOR SALES. IRRADIATION WILL PRESERVE THE APPLE FOR A LONGER SHELF LIFE.

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APPLE IRRADIATION MORE PROBLEMS APPLE IS NOT UNIFORM THOUGH ITS SOLID STATE SKIN OR PEEL IS THICKER SEEDS CORE UNDER PEEL IS DIFFERENT DENSITY THAN NEAR CORE

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Computer Tomography (CT) Slice thickness 1,3,5 mm Cross-sectional resolution 0.2 mm x 0.2 mm CT number Water = 0 Air = -1000

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A slice image of an apple ( 0.9 mm x 0.9 mm)

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Tasks in Monte Carlo Transport Random Sampling Particle Generation Particle Streaming Particle Collisions Geometry Information Particle Interaction Physics Tallies

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A SIMULATION JUST LIKE THE SIMPLE ONE THIS SIMULATION IS RUN BY EITHER PARALLEL COMPUTERS OR A VERY POWERFUL ONE DATA IS GIVEN ON HOW IS THE BEST WAY TO IRRADIATE THE FRUIT

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PRACTICE 1. DESCRIBE HOW THE MONTE CARLO SIMULATION COULD BE USED TO RADIATE CANCER CELLS AND WHY? 2. DESCRIBE HOW THE MONTE CARLO SIMULATION COULD BE USED IN THREE OTHER SITUATIONS AND EXPLAIN.

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GOODBYE THIS SIMULATED CLASSROOM IS NOW OVER!

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