Objectives

Understand how computers can be used to represent real-world phenomena or outcomes
Compare simulations with real-world contexts.
Implement code to mimic real world situations, problems, or phenomena.

What are simulations by College Board definition?

Simulations are abstractions that mimic more complex objects or phenomena from the real world
- Purposes include drawing inferences without the constraints of the real world
Simulations use varying sets of values to reflect the changing state of a real phenomenon
Often, when developing a simulation, it is necessary to remove specific details or simplify aspects
- Simulations can often contain bias based on which details or real-world elements were included/excluded
Simulations allow the formulation of hypotheses under consideration
Variability and randomness of the world is considered using random number generators
Examples: rolling dice, spinners, molecular models, analyze chemicals/reactions...

Analyzing an Example: Air-Traffic Simulator

Say we want to find out what the optimal number of aircrafts that can be in the air in one area is.
A simulation allows us to explore this question without real world contraints of money, time, safety
- Unfortunately we can't just fly 67 planes all at once and see what happens
Since the simulation won't be able to take all variables into control, it may have a bias towards one answer
Will not always have the same result

Functions we often need (python)

import random # a module that defines a series of functions for generating or manipulating random integers
random.choice() #returns a randomly selected element from the specified sequence
random.choice(mylist) # returns random value from list
random.randint(0,10) #randomly selects an integer from given range; range in this case is from 0 to 10
random.random() #will generate a random float between 0.0 to 1.

Functions we often need (js)

// Math.random(); returns a random number
// Math.floor(Math.random() * 10); // Returns a random integer from 0 to 9:

College Board Question 1

Question: The following code simulates the feeding of 4 fish in an aquarium while the owner is on a 5-day trip:

numFish ← 4

foodPerDay ← 20

foodLeft ← 160

daysStarving ← 0

    REPEAT 5 TIMES {

    foodConsumed ← numFish * foodPerDay

    foodLeft ← foodLeft - foodConsumed

    IF (foodLeft < 0) {

        daysStarving ← daysStarving + 1

    }

}

This simulation simplifies a real-world scenario into something that can be modeled in code and executed on a computer.
Summarize how the code works:
- uses the amount of fish, food, and food the fish eat a day to simulate after 5 days how many days will the fish be starving

Examples

Card Flip

import random

cards = ["Ace", "2", "3", "4", "5", "6", "7", "8", "9", "10", "Jack", "Queen", "King"] 
suits = ["Diamonds", "Hearts", "Spades", "Clubs"]

print(random.choice(cards) + " of " + random.choice(suits))

Coin Flip

import random

def coinflip():         #def function 
    randomflip = random.randint(0, 1) #picks either 0 or 1 randomly 
    if randomflip == 0: #assigning 0 to be heads--> if 0 is chosen then it will print, "Heads"
        print("Heads")
    else:
        if randomflip == 1: #assigning 1 to be tails--> if 1 is chosen then it will print, "Tails"
            print("Tails")

def weightedFlip():
    randomflip2 = random.randint(0, 10)
    if(randomflip2 == 0):
        print("Tails")
    else:
        print("Heads")

#Tossing the coin 5 times:
t1 = coinflip()
t2 = coinflip()
t3 = coinflip()
t4 = coinflip()
t5 = coinflip()

print("Weighted Flips: ")

for i in range(5):
    weightedFlip()

Heads
Tails
Tails
Tails
Tails
Weighted Flips: 
Heads
Tails
Heads
Heads
Heads

Your turn: Change the code to make it simulate the flipping of a weighted coin.

Adding images (in Python)

Add a heads and tails images into your images directory with the correct names and run the code below

import random

# importing Image class from PIL package
from PIL import Image
 
# creating a object
im = Image.open(r"images/HeadsOn.png")
image = Image.open(r"images/TailsOn.png")

i=random.randint(0,1)

if i == 1:
    print("heads")
    display(im)

else:
    print("tails")
    display(image)

heads

In order to display an image in python, we can use the PIL package we previously learned about.

Spin the Wheel

import random

from PIL import Image
 
bluePixel = Image.open(r"images/4x4-0000ffff.png")
redPixel = Image.open(r"images/4x4-ff0000ff.png")

print("Spin the wheel!")
print("----------------------------------")

n = 30
blue = 0
red = 0
 
for i in range(n):
    spin = random.randint(1,2)
    if spin == 1: # head
        display(bluePixel)
        blue = blue + 1
    else:         # tail
        display(redPixel)
        red = red + 1
 
print('Number of blue:', blue)
print('Number of red:', red)

Spin the wheel!
----------------------------------

Number of blue: 17
Number of red: 13

Your turn: Add a visual to the simulation!

displays a little pixel for each time red or blue is picked

Population Growth and Plots

import random

totalPopulation = 50 
growthFactor = 1.00005
dayCount = 0 #Every 2 months the population is reported

while totalPopulation < 1000000:
    totalPopulation *= growthFactor
    #Every 56th day, population is reported
    dayCount += 1
    if dayCount % 56 == 0: 
        print("Day " + str(dayCount) + ": " + str(totalPopulation))

Here we initialize the total population to be 50, then set the growth factor as 1.00005 (.005 percent change). It will print the population every 56th day until it reaches one million. It multiplies the current population by the growth factor in each iteration, and increments the day count. When the day count reaches 56, it prints the current population and resets the day count to 0.

Note! This simulation assumes that the growth factor remains constant as time progresses, which may not be a realistic assumption in real-world scenarios.

import matplotlib.pyplot as plt

# Define the initial population and growth rate
population = 100
growth_rate = 0.05

# Define the number of years to simulate
num_years = 50

# Create lists to store the population and year values
populations = [population]
years = [0]

# Simulate population growth for the specified number of years
for year in range(1, num_years+1):
    # Calculate the new population size
    new_population = population + (growth_rate * population)
    # Update the population and year lists
    populations.append(new_population)
    years.append(year)
    # Set the new population as the current population for the next iteration
    population = new_population
    
# Plot the population growth over time
plt.plot(years, populations)
plt.xlabel('Year')
plt.ylabel('Population')
plt.title('Population Growth Simulation')
plt.show()

If we create quantative data, we can plot it using the Matplotlib library.

Example on how simplification can cause bias

import random

beak =  ["small-beak", "long-beak", "medium-beak"],
wing = ["small-wings", "large-wings", "medium-wings"],
height = ["short", "tall","medium"]


naturaldisaster = ["flood", "drought", "fire", "hurricane", "dustbowl"]


print("When a" , random.choice(naturaldisaster) , "hit",  random.choice(height), "birds died")

How does this simulation have bias?

It does not take into account many real-world factors and just does everything by random choice.

JS examples

Dice Roll Binary Coin Flip Card Pull

Hacks

Answer all questions and prompts in the notes (0.2)
Create a simulation
1. Create a simulation that uses iteration and some form of data collection (list, dictionary...) (0.4)
  - try creating quantative data and using the Matplotlib library to display said data
  - Comment and describe function of each parts
  - How does your simulation help solve/mimic a real world problem?
  - Is there any bias in your simulation? Meaning, are there any discrepancies between your program and the real event?
Answer these simulation questions (0.3)
MCQs
- AB
- A
- A
- D
- BD
- C
Bonus: take a real world event and make a pseudocode representation or pseudocode on a flowchart of how you would make a simulation for it (up to +0.1 bonus)

k = -5
m = 2
fs = ma
fs = kx
a = kx/m

x = xi + vit + 1/2at^2
vf = vi + at

import matplotlib.pyplot as plt

k = -5 # spring constant in N/m
m = 2 # mass in kg
t = 0.001 # time step in seconds (smaller is better)
x = 10 # starting position in meters (amplitude)
v = 0 # inital velocity
a = 0 # acceleration

simulationTime = 10 # time for the simulation to run in seconds
steps = simulationTime / t

xs = [] # positions
ts = [] # times

for i in range(int(steps)):
    # kinematics equations
    x = x + v * t + 0.5 * a * pow(t,2)
    v = v + a * t
    # spring force equation
    a = k * x / m
    xs.append(x)
    ts.append(t * i)

plt.plot(ts, xs)
plt.xlabel('Time (seconds)')
plt.ylabel('Position (meters)')
plt.title('Spring simple harmonic motion simulation')
plt.show()

# This simulates the motion of a spring undergoing simple harmonic motion
# This is a nice simulation because it shows a complicated form of motion and translates it into a graph.
# It is also nice because it is much easier to have different starting positions and starting speeds than 
# deriving the equations themselves.
# This simulation gets more inaccurate the furthur it runs and with the bigger the step size. If I made the step size 
# 0.1 seconds it would get very innacurate. The size is small enough here so that it doesn't show much.
# This program also ignores friction, air resistance, and other real-world factors that would affect the motion of the spring