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Normalization, nonnegativity, monotonicity, complements, and differences in a finite probability space
Statement
In a finite probability space , events satisfy and Probability zero need not imply that an event is empty.
Facts & Assumptions
Given: A finite probability space and events .
Event probability is the sum of the nonnegative weights of its outcomes, and the sum of all outcome weights is (Finite probability spaces, outcome weights, events, and event probabilities).
Finite sums preserve nonnegativity and order (Laws of finite sums and finite products).
The real numbers form a totally ordered field (The reals form a totally ordered field).
Proof
The empty sum is , while the sum over is , so and .
Since every summand in is nonnegative, . Splitting the sum over into and gives , so .
Taking in step 1.2 gives ; rearranging the same identity for general gives .
The definition permits zero weights, so a singleton outcome of weight zero is a nonempty event of probability zero. All displayed conclusions follow.
Depends on
Used by
- Two-event inclusion-exclusion: ℙ(A∪ B)=ℙ(A)+ℙ(B)-ℙ(A∩ B) Corollary
- A loaded die as a nonuniform finite probability space Example
- A parameter ledger for the high-girth, high-chromatic alteration proof Example
- The random-colouring proof of R(k,k)>2^k/2 Example
- Mutual independence is inherited by subfamilies and by replacing events with complements Lemma
- An event of positive probability in a finite probability space is nonempty Theorem
- For all positive k,ℓ, some finite graph has girth greater than ℓ and chromatic number greater than k Theorem
- If k≥1 and n≥3k² 2ᵏ, an n-vertex tournament with property Sₖ exists Theorem
- The finite union bound Theorem
- The first-moment method for avoiding or forcing a finite count of bad events Theorem
- The law of total probability for a finite partition Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 52 results over 17 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 1.1 (standard reference, not scraped)
- H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Sections 1.3.2-1.3.3 (standard reference, not scraped)