John Travis
Mississippi College
MATH 353  Introduction to Mathematical Probability and Statistics
Textbook: Tanis and Hogg, A Brief Course in Mathematical Statistics
Basic Concepts
Sets and Probability
Probabilites in general deal with measuring expectation. Later in the course, one will discover natural ways for quantifying these measurements but for now experimenting may help us understand probability a bit better.
Given a set $S$, the corresponding Power Set $P(S)$ is the collection of all subsets of $S$. In the experiment below, the given set $S$ consists of the elements $a, b$ and $c$. The cardinality of a set is the number of elements in that set so for this example, the cardinality should be 8.
Given the sample space {'a', 'c', 'b'} The set P(S) of all subsets is given by {{'a', 'c', 'b'}, {'b'}, {'a'}, {}, {'c', 'b'}, {'a', 'c'}, {'c'}, {'a', 'b'}} Cardinality of P(S) is 8 Given the sample space {'a', 'c', 'b'} The set P(S) of all subsets is given by {{'a', 'c', 'b'}, {'b'}, {'a'}, {}, {'c', 'b'}, {'a', 'c'}, {'c'}, {'a', 'b'}} Cardinality of P(S) is 8 
Above, we saw that three elements give rise to a power set consisting of 8 elements. Experiment with the number of elements in $S$ and make a conjecture regarding the number of elements in $P(S)$ for various sized sets $S$.
Enter any desired elements for your set separating each by a comma. Click on "Update" twice when you are done.
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For other situations, you might have a number of sets who perhaps share elements. A nice visual way to organize the elements is using a Venn Diagram. Below is a Venn diagram application when data is divided between three sets. Play with the elements in each set and see how the resulting Venn Diagram changes.
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Probability and Relative Frequency
Sometimes we can logically determine what the probability for a given outcome should be. At other times, it may be more difficult or impossible to deduce such values. If so, we can often determine an "empirical probability" by using the relative frequency from an experiment.
Coins and Dice
For the simplest example, let's flip some coins. Notice as you change the number of rolls how the relative frequency constantly changes. You will likely even get a different result as you reuse the same number of rolls again and again.
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Notice, since these are random experiments where the likelihood of a given outcome on each trial should not be dependent upon previous trials (the word to use is "independent"), then you could possibly get all of one outcome of another. However, you wouldn't "expect" that possibility for larger values of num_rolls.
Tasks for you:
Make a conjecture regarding the precise value for the actual probability of a each outcome of a coin and again for each outcome of a die. Write a paragraph indicating how the results from your experiment corroborate (provide evidence for justifying) your conjecture.
Click to the left again to hide and once more to show the dynamic interactive window 
Click to the left again to hide and once more to show the dynamic interactive window 
