Dr. J introduces combinations (without replacement) which is an unordered set of k elements taken from a set of n elements where elements cannot be repeated. He provides intuition behind the formula for combinations, which is n factorial divided by (n-k) factorial times k factorial. Namely, combinations (without replacement) are just permutations (without replacement) except without the k factorial permutations of the k items. Thus the formula is the formula for permutations (without replacement) divided by the number of permutations (without replacement) of k items. This formula is applied to find the number of possible starting hands for Texas Hold ‘Em and Euchre.

]]>Permutations (without replacement) are an ordered set of k elements taken from a set of n elements where elements cannot be repeated. In constructing this set, we have n choices for the first element, (n-1) choices for the second element, and so on until we have (n-k+1) choices for the kth element. Thus the formula for the number of permutations without replacement is n factorial divided by (n-k) factorial. This formula is applied to cross-over clinical trials to determine the number of possible treatment plans and to shuffling a deck of cards to determine how many different arrangements of those cards there are.

]]>Dr. J introduces delves into more detail on permutations with replacement including intuition behind the formula and additional examples. To aid the intuition, the Fundamental Rule of Counting is introduced that states “if there are a ways of doing one thing and b ways of doing another thing, then there are a times b ways of doing those two things together.” This is used to “derive” the formula for permutation with replacement (n^k where k is the number of elements in the resulting set and n is the number of elements in the original set). This formula is used to assess the number of possible passwords available in different password policies.

]]>Dr. J introduces the four basic types of counting: combinations and permutations both with and without replacement. Combinations occur when order does not matter while permutations occur when order does matter. The video shows examples of using formulas to perform the counting for each the four basic types of counting.

]]>Dr. J introduces the mathematical definition of probability.

An experiment is any data collection process. A particular result of an experiment is an outcome. The collection of all possible outcomes is the sample space of the experiment. Any subset of the sample space is an event.

Probability is a function of an event. By Kolmogorov’s axioms of probability: 1) this function is a non-negative real number, 2) the probability of the sample space is 1, and 3) the probability of the union of pairwise disjoint set is the sum of their probabilities.

Conditional probability of B given A can be calculated by calculating the probability of the intersection of B and A and dividing by the probability of A.

]]>Dr. J introduces the idea of cardinality, the size of a set. Three categories are discussed: finite, countably infinite, and uncountably infinite. Sets with finite size are those where you can count the elements and eventually stop. Sets with countably infinite size can be put into one-to-one correspondence with the natural numbers. Sets with uncountably infinite size have too many elements to be put into one-to-one correspondence with the natural numbers.

These concepts of set size relate to the mathematical treatment of random variables. Discrete random variables can take on finite or countably infinite number of values while continuous random variables can take on uncountably infinite number of values.

Dr. J introduces a property the set properties of disjoint, pairwise disjoint, and partition. Two sets are *disjoint* if their intersection is the empty set. A collection of sets is *pairwise disjoint* if every pair of sets is disjoint. A collection of sets is a *partition* of S if 1) none of the sets is the empty set (this is not strictly necessary but eliminates trivial situations, 2) the collection is pairwise disjoint, and 3) the union of the sets in the collection is S.

A number of examples are provided as well as Venn Diagrams for each of these properties.

]]>Dr. J introduces Venn Diagrams which are a visual illustration of sets. Venn Diagrams are then used to visually demonstrate subsets and supersets as well as the set operations of union, intersection, set difference, and complement.

]]>Dr. J introduces set operations including union, intersection, set difference and set complement. The union of two sets contains all the elements in either set. The intersection of two sets contains the elements that are in both sets. The set difference of A minus B contains all the elements in A that are not in set B. When working with sets, there is often a universal set S that is the union of all possible sets. The complement of a set A (relative to the universal set S) contains all the elements in S that are not in A. Thus the complement is a set difference, but typically the universal set is implicit.

The video also discusses many properties of these set operations including the identity property of unions and intersections, commutative and associative property of unions and intersections, and the distributive property of unions over intersections and intersections over unions. Additional properties typically referred to as De Morgan’s laws are also introduced.

]]>Dr. J introduces the idea of a subset and set comparisons. Set A is a *subset* of set B if the elements of A are all in the set B. The set A is a *proper* *subset* of B if there is at least one element in B that is not in A. Thus subset is really a *set comparison* and is related to the mathematical inequality *less than* while a *proper subset* is related to the mathematical inequality *less than or equal to*. The opposite set comparison is a *superset* and *proper superset*.