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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:
S₇P₁A₁₂ (containership - area)
S₇P₁A₁₅ (containership - mass)
S₇P₆A₆₁ (grouping/interaction - single-criterion permutation)
S₇P₆A₆₂ (grouping/interaction - multi-criteria permutation)
S₇P₆A₆₃ (grouping/interaction - single-criterion combination)
S₇P₆A₆₄ (grouping/interaction - multi-criteria combination)
S₇P₆A₆₅ (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, S₇P₁A₂₂ (containership - area)
- Force establishes matter in space S₇P₃A₃₂------->S₇P₁A₁₅
S₇P₁A₂₂S₇P₃A₃₂S₇P₁A₁₅... 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.

Blessed are they that have not seen, and yet have believed. John 20:29

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