A mathematician is
performing a probability experiment.
Starting from 0 (zero) on the integer number line, the mathematician
flips a coin and takes one step to the right if it comes up a head or one step
to the left if a tail. This process of flipping and moving either right or left
is repeated for a given number of flips of the coin.
This project computes the probability of ending the walk at
position p after f flips of the coin.
Here is a screenshot reporting
the probability of ending a random walk at +3 after 5 flips as 5/32 instead of
its decimal equivalent 0.15625.
Note that the screen shows
the positions that can be reached for a given number of flips as red-rimmed
black circles. Those that cannot be reached as a function of the number of
flips are represented as white circles.
The number of possible paths to each possible position for a given
number of flips is shown as a number above each possible position. It’s an
interesting fact that these numbers are given by Pascal’s triangle and the nCr
notation for combinations computes
these numbers. The screen also shows that the totals number of possible paths
for a given number of flips is a power of 2.
The probability for ending a
walk at position p after f flips is the number of paths to
position p divided by the total
number of paths for the given number of flips.
This is the link to this
project, Random Walk on the Integer
Number Line Probabilities.
https://scratch.mit.edu/projects/86951083/
A detailed explanation of
the mathematics and the programming used in the project, including a couple of
additional programming simplifications, are detailed in document that is available at no cost
by sending an email request to grandadscience.com.
This Scratch project needed
a power of 2 script, a factorial script, and a nCr script that are available as
separate Scratch projects by clicking on the links as shown below.
Combinations
https://scratch.mit.edu/projects/88028765/
Factorial
https://scratch.mit.edu/projects/87399897/
Powers of 2 Calculator
https://scratch.mit.edu/projects/86877882/
These are the links to my other Random Walk on the Integer Number Line
projects.
Feynman’s
Random Walk
https://scratch.mit.edu/projects/11282377/
Random
Walk with Barriers
https://scratch.mit.edu/projects/11300964/