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SIMULATIONS

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SIMULATIONS

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

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

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

- 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?

- 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.

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

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

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

SETTING UP A SIMULATION ON THE TI-83+

- CHOOSE RANDOM NUMBERS
- DRAW TWO
- RANGE: 0-99
- REPEAT YES
- SET #’S 0-62 AS A POINT. (63 #’s)
- SET #’S 63-99 AS A MISS. (37 #’s)

- 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

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

- COST/DANGER
- NOT MATHEMATICALLY FEASIBLE
- NOT PHYSICALLY FEASIBLE

- 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

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.

AN ADVANCED SIMULATION

MONTE CARLO SIMULATION

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?

color?

What Is The Area Covered By Blue?

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 rectangle2. if it is within the blue area, record this instance a hit3. generate a new location and repeat 10,000 times

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

- 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.

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

- 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.

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

Slice thickness

- 1,3,5 mm
Cross-sectional resolution

0.2 mm x 0.2 mm

CT number

- Water = 0
- Air = -1000

Random Sampling

Particle

Generation

Geometry

Information

Tallies

Particle

Streaming

Particle

Interaction

Physics

Particle

Collisions

- 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

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

THIS SIMULATED CLASSROOM IS NOW OVER!