Probability: Events, Sample Space, Union & Intersection

• Course code: CE-302, EE-302
• The textbook for the course is "Probability and Statistics for Engineering and the Sciences" 9th Edition by Jay L. Devore
• Lecture notes are available on Moodle
• Chapter 2 covers probability
• Section 2.1 is on Sample Spaces and Events (pages 53-57)
• Section 2.2 discusses Axioms, Interpretations, and Properties of Probability (pages 58-66)

Outline of Topics

• Probability Theory
• Random Experiments
• Sample Spaces
• Events
• Set Theory
• Axioms and Properties of Probability

Probability

• Refers to the study of randomness and uncertainty

Probability Theory

• Randomness and uncertainty exist in daily life and in science, engineering, and technology
• Examples include system reliability and communication over unreliable channels

Definitions

• Random experiment: Observing something uncertain like flipping a coin
• Outcome: A result of random experiment (e.g. head)
• Sample space: Set of all possible outcomes (e.g. S = {head, tail})
• Trial: Each repetition of a random experiment
• Event: A collection of one or more outcomes; a subset of the sample space

Random Experiment

• Flipping a fair coin is a random experiment
• The outcome cannot be predicted to be head (H) or tail (T)
• The outcome varies in an unpredictable fashion when the experiment is repeated under the same conditions
• The exact outcomes cannot be determined; instead, their probabilities can be determined

Sample Space

• Sample space, denoted by S, is the set of all possible outcomes of an experiment
• The simplest experiment to which probability applies is one with two possible outcomes
• For example 2.1, when examining a single weld to see whether it is defective, (N represents not defective, D represents defective) and the sample space for this experiment: S = {N, D}
• If a coin were flipped twice the sample space derived from a tree representation is S = {HH,HT,TH,TT}
• Fipping the coin three times leads to a derived sample space of S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}

Example 2.3 with Gas Stations

• Two gas stations are at an intersection, each with six gas pumps
• The random experiment involves counting the number of pumps in use for each station at a specific time
• Possible outcomes cover how many pumps are in use at each station
• With 49 outcomes in S, the sample space is S = {(0, 0), (0, 1), (0, 2), (0, 3), (0, 4),..., (6, 4), (6, 5), (6, 6)}

Events

• An event is any collection (subset C) of outcomes contained in the sample space S
• Simple event: Consists of exactly one outcome: Ex: E₁ = {(0,0)}
• Compound event: Consists of more than one outcome: Ex: C = {(0,0), (0, 1), (1, 0), (1, 1)}

Probability

• Each event A has an assigned number P(A)
• This number P(A) is probability of occurrence of event A
• Probability is a number assigned to an event indicating how "likely" the event will occur during the experiment
• For equally likely events: P(A) = Number of elements of A / Number of elements of S

Set Theory

• Union of two events A and B, denoted by A ∪ B, includes all outcomes that are in A, in B, or in both
• Intersection of two events A and B, denoted by A ∩ B, consists of all outcomes that are in both A and B
• Complement of an event A, denoted by A' (or AC), is the set of all outcomes in S that are not contained in A
• Ø is the null event (event consisting of no outcomes)
• Mutually exclusive events: When A ∩ B = Ø, A and B

Axioms and Properties of Probability

• Axioms (basic properties) of probability:

• Axiom 1: 0 ≤ P(A) ≤ 1

• Axiom 2: P(S) = 1

• Axiom 3: If A ∩ B = Ø, P(AUAB) = P(A) + P(B)

• P(S) = P(AUAC) = P(A) + P(A) = 1 (where A© is the complement of

• P(A©) = 1 - P(A)

• P(Ø) = 0, where Ø is the null event, the event containing no outcomes)

• P(A ∪ B) = P(A) + P(B) - P(A ∩ B)

• P(A ∪ B ∪ C) = P(A) + P(B) + P(C) - P(A ∩ B) - P(A ∩ C) - P(B ∩ C) + P(A ∩ B ∩ C)

Example 2.14

• A residential suburb has the following: 60% of all households get Internet service, 80% get television service, 50% get both services.
• If A = {gets Internet service} then P(A) = 0.6
• If B = {gets TV service} then P(B) = 0.8 and P(A ∩ B) = 0.5
• P(gets at least one of the two services) = P(AUB) = P(A) + P(B) – P(A ∩ B) = 0.6 + 0.8 – 0.5 = 0.9

Example 2.11

• When flipping a coin, either head (H) or tail (T) results
• The sample space is represented therefore by S = {H,T}
• Provided the probability of the event B = {T} is 0.75
• Given A and B disjointed union is S: 1 = P(S) =P(A) + P(B)
• P(A) = 1 – P(B) therefore P(A) = 1 – 0.75 = 0. 25

Determining Probabilities Systematically

• For a finite sample space E1, E2, E3... denote the outcomes with each one consiting of a single event
• Probability of any compound event is computed by ading all probabilities denoted by P(E)'s forall E's in A thus P(A) = ΣP(E¡)
• During off-peak hours, a five-car commuter train provides an example of this

Example 2.15 with Commuter Train

• Probability commuter gets on train car number i being: p = P(car i is selected) = P(E₁)
• If P₁ = P(car 1 is selected) = P(E₁) = 0.1, P2 = P(car 2 is selected) = P(E2) = 0.2, p3 = P(car 3 is selected) = P(E3) = 0.4, P4 = P(car 4 is selected) = P(E4) = 0.2 and p5 = P(car 5 is selected) = P(E5) = 0.1
• and A is the probability of the event that one of the three middle cards is selected
• Then P(A) = ΣP(E₁) = P(E₂) + P(E3) + P(E4) = p2+P3 + P4 = 0.2 + 0.4 +0.2 = 0.8

Equiprobable Outcomes

• For the sample space of a random experiment with equally likely outcomes with the probability for each being the same, p = P(E):
• For event A, with N(A) denoting the number of contained outcomes,
• P(A) = ΣP(E₁) = Σ = where computing probabilities reduces counting
• To do this, the number of N(A) outcomes in A with outcomes in S must both be in S and form its' ratio

15. Example 2.16

• The total sample space of possible outcomes: P(A) = ΣP(E₁) = Σ = ( where computing probabilities reduces counting

Test your knowledge of probability theory with questions on simple vs. compound events, probability representation, sample spaces, and event calculations. Practice set theory concepts like union, intersection, and complement, plus random experiments.