Probability 1: Combinatorial Analysis

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  1. Define combinatorial analysis (or combinatorics)
    Combinatorics is a branch of mathematics that studies countable discrete structures. It includes studying counting, ordering and other problems.
  2. The basic principle of counting
    If event A has m possible outcomes, and event B has n possible outcomes, then there are m*n possible outcomes of the two experiments together.
  3. Number of permutations of n objects
    n*(n-1)*(n-2)*...*3*2*1 = n!
  4. Number of permutations of n objects, of which n1 are alike, n2 are alike, ... nr are alike.
  5. The number of distinct subsets of size k that can be selected from a set of n objects (order of objects is irrelevant)
    , a.k.a. the binomial coefficient.
  6. The number of distinct ordered subsets of size k that can be selected from a set of n objects (order of objects is relevant)
  7. Pascal's Rule (a combinatorial identity about binomial coefficients)
    , for 1 ≤ r ≤ n
  8. The binomial theorem
    • Binomial theorem describes the algebraic expansion of powers of a binomial.
  9. The number of subsets of a set of n elements
    2n   (this includes the null subset)
  10. The number of possible partitions of a set of n objects into r distinct groups (order irrelevant)
     , a.k.a. the multinomial coefficient
  11. The multinomial theorem
    • (the sum is over all non-negative integers n1, n2, ..., nr such that n1+n2+...+n= n)
  12. The number of distinct positive integer-valued vectors  satisfying 
  13. The number of distinct non-negative integer-valued vectors  satisfying 
  14. Type I error
    • A false positive
    • The null hypothesis is rejected when it is actually true.
    • "I falsely think that the Alternative hypothesis is true."
  15. Type II error
    • A false negative
    • The null hypothesis is accepted when it is actually false.
    • "I falsely think that the Alternative hypothesis is false."

Card Set Information

Probability 1: Combinatorial Analysis
2013-05-29 01:07:15
statistics probability

Combinatorial analysis formulas and definitions used in probability theory
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