Expressions Of Pj Problems

Pj Problems - Overview

Celestial Stars

The Number Line

Geometries

7 Spaces Of Interest - Overview

Triadic Unit Mesh

Creation

The Atom

Survival

Energy

Light

Heat

Sound

Music

Language

Stories

Work

States Of Matter

Buoyancy

Nuclear Reactions

Molecular Shapes

Electron Configurations

Chemical Bonds

Energy Conversion

Chemical Reactions

Electromagnetism

Continuity

Growth

Human-cells

Proteins

Nucleic Acids

COHN - Natures Engineering Of The Human Body

The Human-Body Systems

Vision

Walking

Behaviors

Sensors Sensings

Beauty

Faith, Love, Charity

Photosynthesis

Weather

Systems

Algorithms

Tools

Networks

Search

Differential Calculus

Antiderivative

Integral Calculus

Economies

Inflation

Markets

Money Supply

Painting

Venn Diagram - The PjProblemStrings Sequences

The Venn Diagram (named after John Venn who invented it), is a basic model generally used to group members of two or more different sets. In particular, it highlights unique groups of sets and the members common to the groups.

A probabilistic view of the Venn Diagram sees the Venn Diagram as an *outcome space* consisting of probable *event spaces* resulting from a *random experiment*. It is this view that is the focus here.

(ai) What is a random experiment? (ii) What is an *outcome space*? (iii) What is an event?

(b) Is the Universe an outcome Space? Explain.

(c) Is an *empy space* an *outcome space*?

(d) Highlight the key aspects of the Venn Diagram

(e) Asssign PjProblemStrings to your highlights in (d). Thus establish the PjProblemString Sequences.

(f) Suppose state X has a northern region (NR), a central region (CR) and a southern region (SR). The following percentages is from X's department of housing:

28 % of X's residents own houses in NR

30 % of X's residents own houses in CR

18 % of X's residents own houses in SR

14 % of X's residents own houses in NR and CR

12 % of X's residents own houses in NR and SR

10 % of X's residents own houses in CR and SR

8 % of X's residents own houses in all three regions

What % of X's residents do not own houses in the state?

(g) The illustrated Venn Diagram focuses on two and three event spaces and their intersections. Can the Venn Diagram be used to illustrate four or more event spaces and their intersections?

(h) Give an example of an outcome space with four or more event spaces that intersect.

**The strings**: all PjProblems at play.
**The math**: primary PjProblems of interests:

S_{7}P_{1}A_{12} (containership - area)

S_{7}P_{1}A_{15} (containership - mass)

S_{7}P_{6}A_{61} (grouping/interaction - single-criterion permutation)

S_{7}P_{6}A_{62} (grouping/interaction - multi-criteria permutation)

S_{7}P_{6}A_{63} (grouping/interaction - single-criterion combination)

S_{7}P_{6}A_{64} (grouping/interaction - multi-criteria combination)

S_{7}P_{6}A_{65} (grouping/interaction - physical)

**(ai)** A *random experiment* is an experiment or trial characterized by short-term unpredictability of its outcomes; and long-term predictability of the relative frequency of its various outcomes after many trials. For example, consider the toss of a coin with two possible outcomes (Head/Tail). A toss of the coin is a trial. The outcome of the toss is not perfectly predictable in one trial. It could be a head, it could be a tail. However, after many tosses a regular pattern begins to emerge. The number of heads relative to the number of tosses (relative frequency) will begin to tend towards a fixed number. Ditto for the number of tails. In a fair coin, the relative frequency is 0.5 for each outcome. In other words, the outcomes of a *random experiment* are unpredictable in the short-term; but predictable in the long-term after many trials. The *long term* may or may not be synonymous with *long-time*. For example, one can carry out many tosses of a coin in a single day but not the accumulation of actual mortality.

(ii) An *outcome space* (usually denoted by S) is the set of all possible outcomes of a given *random experiment*. A non-probabilistic definition of an *outcome space* is: a *space where things can occur without the occurrences being conditioned on random experiments*. For example, a party in a park.

(iii) An *event* is a subset of *outcome space* in the probabilistic sense or just an *occurrence*.
**(b)** Yes. There are *happenings* in it. However, whether the Universe is a consequence of a random experiment or of the executed plans of an intelligent Being, is a matter of one's belief system.
**(c)** No. It only has the potential to become an *outcome space*.
**(d)** The initial condition of the outcome space of a Venn Diagram was at some point in time an empty space.

Then, forces, motions, changes, groupings/interactions established its identity and dynamism as an outcome space.
**(e)** - Initial empty space, S_{7}P_{1}A_{22} (containership - area)

- Force establishes matter in space S_{7}P_{3}A_{32}------->S_{7}P_{1}A_{15}

S_{7}P_{1}A_{22}S_{7}P_{3}A_{32}S_{7}P_{1}A_{15}... This sequence is the basic PjProblemStrings Sequence of all Venn Spaces. The additional PjProblemStrings that extend the sequence depend on the nature of the particular forces, motions, changes and groupings/interactions that established the identity and dynamism of the outcome space of a given Venn Diagram.
**(f)** This is a probability problem. Methods used to solve probability problems are based on probability axioms and theories. The theories of probability are broadly grouped into:

- *The equally likely outomes theory*: each member of an *outcome space* has equal chance of occurence.

- *The relative frequency theory*: chance of occurence of a member of an *outcome space* tends towards a limit after many repeated trials.

- *The subjective theory*: chance of ocurrence is subjectively derived from indirect data that are not results from repeated trials. For example, prediction of an earthquake in space X 30 years into the future is not based on many repeated trials of actual earthquakes.

The axioms of probabilities are broadly grouped into:

- Negative probability is disallowed. In other words, the probability of an event A = P(A) ≥0.

- Probability of the entire *outcome space* is always equal to 100% (1).

- If the events A and B are *disjoint* (mutually exclusive), the probability that either of the events happens is the sum of the probabilities that each happens

So, AB = {} implies P(A U B) = P(A) + P(B)

- Events have complements. For example, the complement of event A = A^{c} and the probability of P(A^{c}) = 1 - P(A).

- The *union* and *intersection* of sets are important probability terms derived from set theory.

The *union* of A and B = A U B = elements in A or in B or in both.

The *intersection* of A and B = A ∩ B = elements common to both A and B.

Now, let A, B, C represent the events of house ownership in X's NR, CR and SR respectively

And let P(A) = probability of event A.

Then, % of X's residents that do not own houses in the state = P[(A U B U C)^{c})]

P[(A U B U C)^{c})] = 1 - P(A U B U C)

1 - P(A U B U C) = 1 -[P(A) + P(B) + P(C) - P(A∩B) - P(A∩C) - P(B∩C) + P(A∩B∩C)]

So, 1 - P(A U B U C) = 1 - [0.28 + 0.30 + 0.18 - 0.14 - 0.12 -0.10 + 0.08]

So, 1 - P(A U B U C) = 1 - [0.48] = 0.52

So, 52% of X's residents do not own houses in the state.
**(g)** Yes. However, the illustration will be visually complex.
**(h)** An extension of problem (f) is one such example. X becomes a country with four or more states or regions. Some residents own houses in only one state and Some residents of X own houses in two or more states.

Math

The *point* **.** is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

Derivation Of The Area Of A Circle, A Sector Of A Circle And A Circular Ring

Derivation Of The Area Of A Trapezoid, A Rectangle And A Triangle

Derivation Of The Area Of An Ellipse

Derivation Of Volume Of A Cylinder

Derivation Of Volume Of A Sphere

Derivation Of Volume Of A Cone

Derivation Of Volume Of A Torus

Derivation Of Volume Of A Paraboloid

Volume Obtained By Revolving The Curve y = x^{2} About The X Axis

Single Variable Functions

Absolute Value Functions

Conics

Real Numbers

Vector Spaces

Equation Of The Ascent Path Of An Airplane

Calculating Capacity Of A Video Adapter Board Memory

Probability Density Functions

Boolean Algebra - Logic Functions

Ordinary Differential Equations (ODEs)

Infinite Sequences And Series

Introduction To Group Theory

Advanced Calculus - Partial Derivatives

Advanced Calculus - General Charateristics Of Partial Differential Equations

Advanced Calculus - Jacobians

Advanced Calculus - Solving PDEs By The Method Of Separation Of Variables

Advanced Calculus - Fourier Series

Advanced Calculus - Multiple Integrals

Production Schedule That Maximizes Profit Given Constraint Equation

Separation Of Variables As Solution Method For Homogeneous Heat Flow Equation

Newton And Fourier Cooling Laws Applied To Heat Flow Boundary Conditions

Fourier Series

Derivation Of Heat Equation For A One-Dimensional Heat Flow

Homogenizing-Non-Homogeneous-Time-Varying-IBVP-Boundary-Condition

The Universe is composed of *matter* and *radiant energy*. *Matter* is any kind of *mass-energy* that moves with velocities less than the velocity of light. *Radiant energy* is any kind of *mass-energy* that moves with the velocity of light.

Periodic Table

Composition And Structure Of Matter

How Matter Gets Composed

How Matter Gets Composed (2)

Molecular Structure Of Matter

Molecular Shapes: Bond Length, Bond Angle

Molecular Shapes: Valence Shell Electron Pair Repulsion

Molecular Shapes: Orbital Hybridization

Molecular Shapes: Sigma Bonds Pi Bonds

Molecular Shapes: Non ABn Molecules

Molecular Orbital Theory

More Pj Problem Strings