Unit 2 of 13
Counting and Equally Likely Models
- 7 lessons
- Videos 9 to 15
Counting gives probabilities when outcomes are equally likely. This unit develops the sum and product rules, permutations, combinations, multinomial counts, inclusion-exclusion and collision problems.
Each method starts from what one outcome records, whether order matters, whether repetition is allowed and which objects can be told apart. Combinations remove the overcount of ordered selections, and repeated labels are handled by dividing out their own arrangements. Inclusion-exclusion corrects for the overlaps of three events, and the birthday problem is solved through its no-collision complement. The last lesson shows why erasing order can leave outcomes that are no longer equally likely.
Every lesson will be one video. The videos are coming soon, and each lesson page already has the definitions and an example for every key term.
Ordered and Unordered Selections
Sum and Product Rules for Counting
How the product rule counts outcomes built in stages and the sum rule counts disjoint alternatives, with cases, decision trees, selection with and without replacement, and restrictions.
Video 9
Permutations
What a permutation is, how factorials count arrangements of distinct objects, why zero factorial equals 1, how to count partial permutations, and what changes for a circular arrangement.
Video 10
Combinations and Binomial Coefficients
How combinations count unordered selections by removing the overcount of ordered ones, giving the binomial coefficient, its symmetry, Pascal's identity and triangle, and the binomial theorem.
Video 11
Repeated Objects and Multinomial Counting
How to count arrangements when labels repeat, why the multinomial coefficient divides by the shuffles of identical objects, and how it counts allocations to named groups and gives the coefficients of a multinomial expansion.
Video 12
Overlaps and Collisions
Inclusion-Exclusion
How inclusion-exclusion corrects a sum of three event probabilities for overlaps, why pairwise intersections contain the triple intersection, and how disjoint Venn regions give exactly one or exactly two of three events.
Video 13
Birthday and Collision Problems
How the birthday problem and other collision probabilities for independent uniform choices follow from counting no-collision records, and how they differ from a specified pair, the pigeonhole case and unequal categories.
Video 14
Choosing a Counting Method
Choosing a Counting Method
How to choose between ordered selections, subsets and multisets from what one outcome records, how the multiplicity of a record decides whether erasing order keeps outcomes equally likely, and when to count by complement.
Video 15