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.

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.

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.

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.