ANOP Week 5

Uncertainty is a fundamental aspect of decision-making.
Significant effort is invested in planning for and reacting to uncertainty.
Probability is a numerical measure illustrating the likelihood of an event occurring.
This measure is typically conveyed through a probability distribution.
Probability distributions assist decision-makers in evaluating potential actions and determining the optimal course.
The foundational elements for studying probability are outcomes and events.
An event is formally defined as a collection of outcomes.

A random experiment is a process that generates clearly defined results or outcomes.
The sample space (S) for a random experiment is identified by enumerating all its possible outcomes.
Example 1: Tossing a Coin
- Experimental outcomes (sample points): Head, Tail.
- Sample space: S = { \text{Head, Tail} }.
Example 2: Rolling a Die
- Experimental outcomes (sample points): 1, 2, 3, 4, 5, 6.
- Sample space: S = { 1, 2, 3, 4, 5, 6 }.
Defining Events for Rolling a Die:
- Let A be the event of getting number 5: A = { 5 }.
- Let B be the event of getting a number at most 5: B = { 1, 2, 3, 4, 5 }.
- Let C be the event of getting 1: C = { 1 }.
- Let D be the event of getting no larger than 3: D = { 1, 2, 3 }.
- An event is considered to occur if any one of its constituent outcomes occurs.

Probability is a numerical value, p, which must satisfy 0 \le p \le 1.
A probability of 1 signifies that the event is certain to occur.
A higher probability indicates a greater likelihood of the event occurring.
The probability of an event is calculated as the sum of the probabilities of all outcomes that constitute the event.
Example: Rolling a Die (Event A)
- A = { 5 } (getting number 5).
- P(A) = P(\text{get number 5}) = P(5) = \frac{1}{6}.
Example: Rolling a Die (Event D)
- D = { 1, 2, 3 } (getting no larger than 3).
- P(D) = P(1) + P(2) + P(3) = \frac{1}{6} + \frac{1}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}.
Example: CP&L Capacity Expansion Project
- Management expects project completion in 8, 9, 10, 11, or 12 months based on 40 past projects.
- Sample Space: All possible outcomes of completion times.
- Probabilities of Outcomes:
  Completion Time (months)
  No. of Past Projects
  Probability of Outcome
  8
  6
  6/40 = 0.15
  9
  10
  10/40 = 0.25
  10
  12
  12/40 = 0.30
  11
  6
  6/40 = 0.15
  12
  6
  6/40 = 0.15
  Total
  40
  1.00
- Event C: Project completed in 10 months or less.
  - C = { 8, 9, 10 }.
  - P(C) = P(8) + P(9) + P(10) = 0.15 + 0.25 + 0.30 = 0.70.
- Event L: Project completed in less than 10 months.
  - L = { 8, 9 }.
  - P(L) = P(8) + P(9) = 0.15 + 0.25 = 0.40.
- Event M: Project completed in more than 10 months.
  - M = { 11, 12 }.
  - P(M) = P(11) + P(12) = 0.15 + 0.15 = 0.30.

The complement of event A (denoted by A^C) is the event composed of all outcomes that are not in A.
In any probability scenario, either event A or its complement A^C must occur.
Formula: P(A) + P(A^C) = 1
Rearranged Formula: P(A) = 1 - P(A^C).
Example: Rolling a Die
- Sample space S = { 1, 2, 3, 4, 5, 6 }.
- Let A be the event of getting number 5: A = { 5 } (P(A) = 1/6).
- The complement A^C is the event of not getting number 5: A^C = { 1, 2, 3, 4, 6 }. (P(A^C) = 5/6).