Probability: Random Experiments and Events

• Chapter 6 focuses on probability.

Random Experiment

• Any process that leads to observing or measuring a certain phenomenon.
• It is an experiment or process with outcomes that cannot be predicted with certainty.
• It is an experiment where the outcomes are known, but which specific outcome will occur is unknown.

Sample Space

• Denoted by S, represents the set of all possible outcomes of a random experiment..
• Sample spaces can be discrete or continuous.

Event

• Any subset of the sample space.

Types of Events

• Sure Event: Contains all elements of the sample space, where A= S.
• Impossible Event: An empty set with no outcomes, where A= { } or Ø.
• Simple Event: A single point.
• Compound Event: Consists of more than one single point.

Events

• Independed Event: If any one event is unaffected by any other.
• Depended Event: If any one event is affected by any other.
• Complement Event: Contains all elements belonging to S, but not to A.
• A ∩ A^c = {} = 0, A ∪ A^c = S.
• Disjoint Event (Mutually Exclusive): The occurrence of one event prevents the occurrence of the other, noted as (A ∩ B)= Ø.
• Non Disjoint (Non-Mutually Exclusive):The occurrences of one event does not preclude the other.
• Exhaustive Events: If the union of events results in the universal set which is the sample space.
• Equally Likely Outcomes: Outcomes of a random experiment are considered equally likely if each outcome has the same chance (probability) of occurring.

Key Probability Concepts and Examples

• When flipping a fair coin, the possible outcomes are a head (H) and a tail (T), both equally likely

• Hence, P(H) = n(H)/n(S) = 1/2 and P(T) = n(T)/n(S) = 1/2.

• For the number set 1,2,3,4,5,6:

• The probability of getting an even number is P(A) = n(A)/n(S) = 3/6 = 0.5, where A = {2,4,6}.

• The probability of getting a number less than 4 is P(B) = n(B)/n(S) = 3/6 = 0.5, where B = {1,2,3}.

• The probability of getting 1 or 3 is P(C) = n(C)/n(S) = 2/6 = 0.33, where C {1,3}.

• The probability of getting an odd number is P(D) = n(D)/n(S) = 3/6 = 0.5, where D = {1,3,5}.

• In a hospital, there are 630 patients classified by blood type, and probabilities are calculated for selecting specific blood types.

Addition Rule

• P(A ∪ B) = P(A) + P(B) – P(A ∩ B)
• An event A occurs with probability 0.5 and an event B occurs with probability 0.4. The probability that both A and B occur is 0.1. The probability of A or B is P(A ∪ B) = 0.5 + 0.4 - 0.1 = 0.8.

Disjoint Events

• For disjoint events (A ∩ B) = { }, and thus n(A ∩ B) = 0, implying P(A ∩ B) = 0.
• Therefore, P(A ∪ B) = P(A) + P(B).
• If A and B are two disjoint events, with P(A) = 0.35 and P(B) = 0.15, then P(A ∪ B) = 0.35 + 0.15 = 0.5.

Independent Events

• P(A∩B)=P(A)×P(B)
• If A and B are two independent events with P(A) = 0.8 and P(B) = 0.7, then P(A ∩ B) = 0.8 * 0.7 = 0.56.
• If P(A ∩ B) = P(A) × P(B), then the events are called independent.

Complement Event (AC)

• A ∩ A^c = {}, and A ∪ A^c = S.

Conditional Probability

• P(B|A) = P(A ∩ B) / P(A), where P(A) ≠ 0
• P(A|B) = P(A ∩ B) / P(B), where P(B) ≠ 0
• Example: In a classification of 400 patients based on smoking and cancer status, probabilities are calculated.

Explore probability with random experiments, sample spaces, and event types. Understand sure, impossible, simple, and compound events. Differentiate between independent, dependent, and complement events. Chapter 6 focuses on probability.