Today's lab continues our work on simulations, focusing on stock simulations.

### Random Walks and Stock Prices

Random walks (and their more sophisticated cousins, Monte Carlo methods) are used to simulate diverse phenomena from predicting stock prices to predicting the weather.

We will focus on simulating how the price of a stock changes over a quarter. We will start with the price of \$0, and each day with probability prob, the price will increase by a dollar. If it does not increase, it's price will go down by a dollar:

```    money = 1

for day in range(90):
if (random() < prob):
money = money + 1
else:
money = money - 1
```
Instead of just tracking one stock, we'll track 10 different ones, using a list, prob[], to hold the probability of change for each stock. We can make a graphic version of this by plotting the change in money made each day:
```def simulate(turtleList, prob, money, time):
#Calculate and graph amount money made over time days
for day in range(time):
for i in range(len(turtleList)):
#If the random number is below the turtle's prob, go up
if (random() < prob[i]):
money[i] = money[i] + 1
turtleList[i].left(45)
turtleList[i].forward(10)
turtleList[i].right(45)
#Else, go down
else:
money[i] = money[i] - 1
turtleList[i].right(45)
turtleList[i].forward(10)
turtleList[i].left(45)

```
To run this program, we need to set up the lists as well as the graphics environment. First, let's write out the main:
```def main():
#Keep track of the money made by each turtle/chain:
money = *10
prob = *10

#Set up screen and turtles
setUpScreen()
turtleList = setUpTurtles(10)
for i in range(10):
prob[i] = random()
print(prob[i])

#Simulate the money gained and lost for each stock over 90 days:
simulate(turtleList, prob, money, 90)

turtle.Screen().exitonclick()

main()
```
We start by setting up empty lists to hold the money made as well as the probability of success for each turtle. Our main() also includes several calls to functions that we need to write:
• setUpScreen(): draws an axis to the screen
• setUpTurtles(): sets up 10 different turtles, all different colors
• simulate(): simulates the amount of money made over the time period

Let's look at each one separately:

setUpScreen(): Creates a turtle just for drawing the x-axis. It lifts the pen, moves far to the left, and then draws across the screen.

```#A function that sets up the x-axis on the screen:
def setUpScreen():
#Create a turtle to draw the x-axis
t = turtle.Turtle()
t.speed(0)
#Lift it's pen up and move it to the far left
t.up()
t.right(180)
t.forward(350)
#Draw the axis
t.down()
t.right(180)
t.forward(700)
```
setUpTurtles(): This function uses a for-loop to create 10 turtles. Each turtle is moved to the starting position (at the far left of the screen), has had its speed adjusted to full (10), and been set to a random color. It is then appended to the end of the list. After all 10 turtles are created, the list is returned.
```#Sets up num turtles to all start at the far left,
#   to tell them apart, color them different shades
def setUpTurtles(num):
turtleList = []             #Our list of turtles
for i in range(num):
t = turtle.Turtle()
t.speed(0)              #Fastest turtle speed to avoid boredom
t.up()
t.right(180)
t.forward(350)
t.right(180)
t.down()
#Choose random values for the red, green, and blue (RGB):
t.color(random(), random(), random())
turtleList.append(t)

return(turtleList)
```
simulate(): This is where the simulation occurs. As in our random walk lab, a random number is generated. If it is less than the probability for the given turtle, the turtle moves up (makes money. Otherwise, the turtle moves down (loses money).
```def simulate(turtleList, prob, money, time):
#Calculate and graph amount money made over 90 days
for day in range(time):
for i in range(len(turtleList)):
#If the random number is below the turtle's prob, go up
if (random() < prob[i]):
money[i] = money[i] + 1
turtleList[i].left(45)
turtleList[i].forward(10)
turtleList[i].right(45)
#Else, go down
else:
money[i] = money[i] - 1
turtleList[i].right(45)
turtleList[i].forward(10)
turtleList[i].left(45)

```

### The Complete Program:

Try running the complete program several times to see how the simulation changes:
```import turtle
from random import *

#A function that sets up the x-axis on the screen:
def setUpScreen():
#Create a turtle to draw the x-axis
t = turtle.Turtle()
t.speed(0)
#Lift it's pen up and move it to the far left
t.up()
t.right(180)
t.forward(350)
#Draw the axis
t.down()
t.right(180)
t.forward(700)

#Sets up num turtles to all start at the far left,
#   to tell them apart, color them different shades
def setUpTurtles(num):
turtleList = []             #Our list of turtles
for i in range(num):
t = turtle.Turtle()
t.speed(0)              #Fastest turtle speed to avoid boredom
t.up()
t.right(180)
t.forward(350)
t.right(180)
t.down()
#Choose random values for the red, green, and blue (RGB):
t.color(random(), random(), random())
turtleList.append(t)

return(turtleList)

def simulate(turtleList, prob, money, time):
#Calculate and graph amount money made over time days
for day in range(time):
for i in range(len(turtleList)):
#If the random number is below the turtle's prob, go up
if (random() < prob[i]):
money[i] = money[i] + 1
turtleList[i].left(45)
turtleList[i].forward(10)
turtleList[i].right(45)
#Else, go down
else:
money[i] = money[i] - 1
turtleList[i].right(45)
turtleList[i].forward(10)
turtleList[i].left(45)

def main():
#Keep track of the money made by each turtle/chain:
money = *10
prob = *10

#Set up screen and turtles
setUpScreen()
turtleList = setUpTurtles(10)
for i in range(10):
prob[i] = random()
print(prob[i])

#Simulate the money gained and lost for each stock over 90 days:
simulate(turtleList, prob, money, 90)