Showing posts with label combinations. Show all posts
Showing posts with label combinations. Show all posts

Friday, December 4, 2015

Random Walk Probabilities

   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/

Wednesday, November 18, 2015

Combinations - nCr


   In the previous post on Factorials I mentioned that I needed a factorial algorithm for a Combinations project and the Combinations project for a Random Walker project.
   As you can see in the formula for the number of combinations of n things taken r at a time with no duplications there are three factorials to compute to get to the nCr.
   This is a screenshot of the Combinations project. The graphic of the fruit is to illustrate that if n = 4 (the pear, orange, banana, and strawberry) then there are four combinations (shown in the four rows) when taken 3 at a time. The project first asks for n, then r and then computes nCr.
   The project can be viewed and downloaded by clicking on this link.
https://scratch.mit.edu/projects/88028765/
   A free document in PDF format describing the coding in more detail can be obtained on request. Send an email to grandadscience@gmail.com.