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TYPOLOGY OF OUR EDUCATIONAL MODELS

Probability and Statistics

MathKit 6.1 is supplied with a new set of tools designed for modeling random experiments and processing statistical data. These tools can be used to:

  • set up random experiments with given conditions of different types;
  • visualize the results of the experiment in real time;
  • run automatically series of independent trials;
  • monitor the change of random variables; in particular, by plotting graphs;
  • collect experimental data in a table;
  • apply statistical methods to process these data and represent the results by various graphs and charts.

6.1. Simple Random Experiments

MathKit tools make it possible to discover interesting patterns even in the most elementary random events. Everybody knows that in a series of coin tosses, the frequency of heads (or tails) approaches 1/2. But how fast? To answer this question, you must conduct hundreds of experiments. The MathKit random player easily performs any desired number of single tosses. Their results can be arranged in a table, and then processed using suitable statistical functions and graphs.

Statistical processing of coin flipping yields the following result: the deviation of the frequency from 1/2 is inversely proportional, in a statistical sense, to the square root of the number of tosses.

Changing Frequency
in Coin Tossing Experiment


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6.2. Random Variable

In the previous section we showed how MathKit models random trials and helps to study the frequencies of their results. In many problems of probability and statistics we have to deal with random variables, numeric functions that depend on the outcome of a random experiment. For example, when we roll two dice, we may want to consider the following random variables:

  • the total of the dice;
  • their product;
  • the maximum of the two values;
  • the minimum of the two values, etc.

On the one hand, these values are random indeed: no one can definitely predict their numerical values. But on the other hand, their behavior is subject to certain regularities. We can discover them using MathKit. In the model below you can see the laws of distribution of these values, calculate their average values and variance.

Random Variables
in the Experiment with Two Dice


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6.3. Random Walk

Another interesting example of a stochastic model associated with many practical problems is random walk on the integer grid on a line, in the plane, or in space.

Imagine a gambler who came to a casino with a Euro and wants to raise his capital to b Euro. What are his chances for success? How long will he have to play before he goes totally broke or achieves his goal? The model below performs random walk on a straight line. It helps to answer these and many other interesting questions.

Gambler's Ruin Problem


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6.4. Geometric Probability

All the models considered above are based on what is called the classical definition of probability, which is used to describe experiments with a finite number of equally likely outcomes. But in many situations the number of outcomes is infinite: such are, for example, the hit point of a dart, or the time when the next customer appears in a supermarket. In this case, the geometric definition of probability, or just geometric probability is used. It is based on a random selection of points from a given interval or region; "equal likeness" of outcomes is reflected by the assumption that the probability to hit a certain region is proportional to its area.

Using geometric probability, we can obtain many interesting results, one of which belongs to the French philosopher and mathematician Georges de Buffon. More than three hundred years ago he discovered that when a needle is dropped on ruled paper, then the probability that it crosses a line is expressed in terms of the number pi, and so the value of pi can be found approximately from the frequency of crossings. MathKit allows not only to reproduce Buffon's Needle experiment, but also to generalize it by using an arbitrary curve instead of a "needle".

Buffon's Needle


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6.5. Probabilistic Games

Another interesting class of stochastic models comprises probabilistic games. With MathKit, you can create them yourself or use off-the-shelf models. In order to win in the most interesting of them, you need not only some luck, but also some knowledge of probability, the ability to choose the right strategy and to calculate your chances for success.

Examples of such games include the so-called "Nontransitive Dice" or the famous Monty Hall Problem about winning one of three items: a car and two goats.

Nontransitive Dice


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Two Goats and a Car


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6.6. How to Use

The tools and models considered in this section can be used in different ways:

  • to demonstrate basic statistical regularities;
  • to make clear statements of probability problems;
  • to verify theoretical results experimentally;
  • to build your own models;
  • for independent experimental research and creating new tasks.

A student may have to work with an off-the-shelf model selected from our collection or constructed by the teacher in advance, or to create his or her own model from scratch.

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