Homework 4, Problem 3: Pi from
pie
[30
points; individual or pair]
Submission: Submit your hw4pr3.py file to Canvas
It is perhaps surprising that
it is possible to estimate the mathematical constant π without resorting
to any techniques or operations more sophisticated than counting, adding, and
multiplication. This problem asks you to write two functions that estimate pi (3.14159...)
by dart-throwing.
The basic idea
Imagine a circle inscribed
within a square that spans the area where -1 ≤ x ≤ 1 and -1 ≤ y ≤ 1. The area of the inscribed circle, whose radius is 1.0 would be π .
If you were to throw darts at
random locations in the square, only some of them would hit the circle
inscribed within it. The ratio
area of the circle / area
of the square
can be estimated by the ratio
number of darts that hit
the circle / total number of darts thrown
As the number of darts
increases, the second ratio, above, gets closer and closer to the first ratio.
Since three of the four quantities involved are known, they can be used to
approximate the area of the circle - this in turn can be used to approximate π
Functions to write
For this problem, you will
write two functions (each based on loops) that use different tests for
determining how many random darts to throw: forPi( n )and whilePi( error ).
The forPi( n ) function
takes in a positive integer n and should "throw" n darts at the
square. Each time a dart is thrown, the function should print
After n throws, the final resulting estimate of π should
be returned.
The whilePi( error ) function, on the other hand, will take in a positive
floating-point value, error, and should then continue to throw darts at the
dartboard (the square) until the resulting estimate of π is less than error. As with the forPi function, this
function should print
after each dart throw it makes.
Also, it should return the number of darts thrown in order to reach the
input accuracy.
Note that the whilePi function requires the actual, known value of π in order to determine whether or not its estimate is
within the error range. Although this would not be available for estimating an
unknown constant, you may include the line
import math
in your code and then use the
value of math.pi as the actual value of π .
Design, or to throw
darts at Python.
This problem allows for any
design of the dart-throwing simulation that you would like to use -- as long as
it is loop-based and not recursion-based (simply for the practice).
To throw a dart, you will want to generate random x and y coordinates
between -1.0 and 1.0. Be
sure to include the line
import random
near the top of your file. When
you do this, you will now be able to use the function
random.uniform( -1.0, 1.0
)
that will return a
floating-point value that is in the range from -1.0 to 1.0
You may want a
helper function that
Such a helper function would
help in both cases: forPi and whilePi.
Write-up of your π
estimator
However you design your Monte
Carlo simulation, you should be sure to include an explanatory docstring for
each of your functions. In addition, if there are any parts of your code that
warrant additional commenting, you should use the in-line comments that begin
with #.
Examples
Here is an example run from each of the two functions -- since there are random
numbers involved, they are here to give a sense of one appropriate way to
handle the output. Each run will be different.
>>> forPi( 10 )
1 hits out of 1 throws so that pi is 4.0
2 hits out of 2 throws so that pi is 4.0
3 hits out of 3 throws so that pi is 4.0
4 hits out of 4 throws so that pi is 4.0
4 hits out of 5 throws so that pi is 3.2
5 hits out of 6 throws so that pi is 3.33333333333
6 hits out of 7 throws so that pi is 3.42857142857
6 hits out of 8 throws so that pi is 3.0
7 hits out of 9 throws so that pi is 3.11111111111
8 hits out of 10 throws so that pi is 3.2
3.2000000000000002
>>> whilePi( 0.1 )
1 hits out of 1 throws so that pi is 4.0
2 hits out of 2 throws so that pi is 4.0
3 hits out of 3 throws so that pi is 4.0
4 hits out of 4 throws so that pi is 4.0
5 hits out of 5 throws so that pi is 4.0
5 hits out of 6 throws so that pi is 3.33333333333
6 hits out of 7 throws so that pi is 3.42857142857
7 hits out of 8 throws so that pi is 3.5
7 hits out of 9 throws so that pi is 3.11111111111
9
If
you have gotten to this point, you have completed problem3. You should submit your hw4pr3.py file at Canvas .
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