# Math Teacher's Resource

## Innovative software and content for today's math teacher

### Blog # PROBABILITY SIMULATIONS SOFTWARE

The Probability Simulations Software is a free tool for teaching core concepts of statistics and probability theory. The software is designed for interactive and dynamic classroom demonstration and experimentation to help your students learn on a deeper level. Graphic outputs from the software also offer teachers an easy way to create custom instructional content. Each software module presented below includes examples of instructional applications and user interface graphics for quick familiarization.

## EXPLORE THE CLASSIC SIMULATIONS MODULE

This module includes the following classic simulations:

▶  Birthday

▶  Casino Craps

▶  Monte Carlo π/4

▶  Buffon's Coin

▶  Buffon's Needle

▶  Triangle and Stick

▶  Numerical Expression

▶  Obtuse Triangle (3 points)

▶  Obtuse Triangle (3 angles)

Each of the classic simulations can be run in manual or automatic mode to demonstrate how the experimental probability converges toward the expected probability as the number of trials increases.  The following examples of software output demonstrate two of the classic simulations, Casino Craps and Buffon's Coin.

• ##### ▷ Casino Craps Simulation

Manual Mode

The  animation to the right shows the simulation of  40 games of Casino Craps being played one game at a time in manual mode. Users can manually increment the simulation count indefinitely at a pace that is appropriate for the lesson being taught.

Game rules are as follows:

• If the shooter rolls 7 or 11 on the come-out-roll (i.e. the first roll of a game), the shooter wins and the game ends.
• If the shooter rolls 2, 3, or 12 on the come-out-roll, the shooter loses the game ends.
• If the shooter rolls 4, 5, 6, 8, 9 or 10 on the come-out-roll, the shooter's roll becomes his point and continues rolling the dice until he wins by rolling his point or loses by rolling a 7.

Automatic Mode

In automatic mode, users can adjust number of experimental trials in the "Edit Parameters" menu to explore how the experimental probability of winning a game converges to the expected probability as the number of games played increases.

• ##### ▷ Buffon's Coin Simulation

Manual Mode

Buffon's coin problem asks to find the probability that a randomly tossed coin with radius r  will intersect a line if given a floor covered with squares or equilateral triangles. This example uses a coin with a radius of 1 unit and a square with side lengths of 10 units. Note that the inside square represents the area where a coin cannot possibly intersect the outer square. The animation below shows the simulation of 35 individual coin tosses in manual mode.

Users can manually increment the simulation count indefinitely at a pace that is appropriate for the lesson being taught.

Automatic Mode

The examples below show results for 1,000 and 10,000 coin tosses in automatic mode.  Users can adjust number of experimental trials in the "Edit Parameters" menu.    ## EXPLORE THE PROBABILITY DISTRIBUTION SIMULATIONS MODULE

This module includes the following probability distribution simulations:

▶  Toss Single Die

▶  Toss N Die

▶  Binomial

▶  Poisson

▶  Geometric

▶  HyperGeometric

▶  Urn with Balls

▶  Normal

▶  Exponential

▶  t-Distribution

▶  F-Distribution

▶  Chi-Square

▶  Uniform Continuous

Each of the probability distributions simulations can be run in manual or automatic mode. Users can adjust the sample population, adjust the confidence interval, conduct hypothesis testing, and experiment with sampling statistics and variance. Cumulative probability, inverse probability, and critical values for various distributions can also be calculated using this module. The following examples of software output demonstrate two of the probability distributions simulations, Binomial and Normal.

• ##### ▷ Binomial Probability Distribution

Manual Mode

The  animation below shows the cumulative results of 70 simulations run in manual mode.  A Bernoulli trial is any experiment that has only two outcomes named "success" and "failure" (e.g. a multiple choice test where each question has only one correct answer). Binomial distributions are used to find the probability of obtaining x successes in n independent Bernoulli trials. In this example, each simulation has 40 Bernoulli trials where the probability for success is 0.3. Users can manually increment the simulation count indefinitely at a pace that is appropriate for the lesson being taught.

Automatic Mode

The graphic below shows the cumulative results of 10,000 simulations run in automatic mode without changing the parameters. Users can adjust number of experimental trials in the "Edit Parameters" menu. • ##### ▷ Normal Probability Distribution

Manual Mode

The normal distribution is important because a wide variety of populations and sample statistics have a normal probability distribution. Random variable x of a normal population is continuous and ranges from -∞ to +∞. The area under the normal curve from µ -kσ  to µ +kσ is always the same for any choice of µ and σ. The  animation below shows interval testing at the 95% confidence level in manual mode for 25 simulations where µ = 0 and σ =1.

Users can manually increment the simulation count indefinitely at a pace that is appropriate for the lesson being taught.

The  animation below shows hypothesis testing at the 5% significance level in manual mode for 25 simulations where µ = 0 and σ =1.

Automatic Mode

The graphic below shows the cumulative probability of 2,000 simulations run in automatic mode with mean µ = 0 and standard deviation σ =1.  Users can adjust number of experimental trials in the "Edit Parameters" menu. Version 3.2.1

Released 8/28/16

(1.47 MB file size)

Download the Probability Simulations Software installation package by clicking on the icon to the right, and install the software on your computer or external hard drive by executing the installer package. This software is FREE - no strings attached. We are excited to see how creative math teachers everywhere implement the software to generate custom instructional content and present  mathematical concepts in their classrooms. Note that the software is licensed for educational purposes only. Please review the License Agreement before downloading and using the software.

• The software requires Microsoft® Vista, 7, 8, or 10 operating system. Sorry, Mac® OS is not supported at this time.
• The software license does not expire.
• Use with confidence; our software contains no advertising, malware, spyware, or other subversive code.
• Technical support is provided for questions regarding use and functionality of the software. now

We hope that you thoroughly enjoy using our software. If you find  the Probability Simulations Software useful, consider making a donation. We are constantly working to improve its functionality and usability while developing new software. We are 100% supported by donations, so your support goes further than you may realize. Thanks so much!

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\$10 suggested donation amount per installation  