Probability Rules and Concepts (Chapter 5, Section 5.1)

Probability

Introduction to Probability

Probability is a fundamental concept in statistics that quantifies the likelihood of events occurring in a random experiment. It provides a mathematical framework for analyzing uncertainty and making predictions about outcomes.

• Probability is a measure associated with how certain we are of outcomes of a particular experiment or activity.

• Experiment: A planned operation carried out under controlled conditions. If the result is not predetermined, it is called a chance experiment.

• Outcome: The result of an experiment.

• Sample Space (S): The set of all possible outcomes of an experiment, usually denoted by an uppercase letter.

• Event: Any combination of outcomes, often denoted by uppercase letters such as A or B.

Example: If you flip one fair coin, the sample space is S = {H, T}, where H = heads and T = tails.

Probability Rules

Probability values are assigned to events based on their likelihood. These values follow specific rules and conventions.

• The probability of an event A is written as P(A).

• Relative frequency is used to estimate the probability of an outcome.

• Probabilities are always between zero and one, inclusive: $0 \\leq P(A) \\leq 1$

• P(A) = 0 means event A can never happen.

• P(A) = 1 means event A always happens.

• P(A) = 0.5 means event A is equally likely to occur or not to occur.

• Equally likely means each outcome of an experiment occurs with equal probability.

Example: Tossing a fair coin, both Head (H) and Tail (T) are equally likely.

Calculating Probability for Equally Likely Outcomes

When all outcomes in the sample space are equally likely, probability is calculated by counting favorable outcomes and dividing by the total number of outcomes.

• Count the number of outcomes for event A.

• Divide by the total number of outcomes in the sample space.

Example: Tossing a fair dime and a fair nickel, sample space is {HH, TH, HT, TT}. Let A = getting one head. Outcomes: {HT, TH}. $P(A) = \\frac{2}{4} = 0.5$

• The sum of the probabilities of all outcomes must equal 1.

• An unusual event is one with a low probability of occurring.

Law of Large Numbers

The law of large numbers is a key principle in probability theory. It states that as the number of repetitions of an experiment increases, the observed relative frequency of an event approaches its theoretical probability.

• Long-term observed relative frequency will approximate the theoretical probability.

• The term empirical is often used for observed probability.

Empirical Method

The empirical method uses observed data to estimate probabilities. It is based on the relative frequency of an event occurring in repeated trials.

• Probabilities are computed using empirical evidence from experiments.

• The probability of event E is approximately: $P(E) = \\frac{\\text{frequency of E}}{\\text{number of trials of experiment}}$

Classical Method

The classical method calculates probabilities without performing experiments, relying on counting techniques and the assumption of equally likely outcomes.

• If an experiment has n equally likely outcomes and event E can occur in m ways, then: $P(E) = \\frac{m}{n}$

• If S is the sample space and N(E) is the number of outcomes in E: $P(E) = \\frac{N(E)}{N(S)}$

"Or" and "And" Events

Events can be combined using union (OR) and intersection (AND) operations.

• Union (OR) Event: The event A OR B includes outcomes in A, in B, or in both. Example: If A = {1,2,3,4,5} and B = {4,5,6,7,8}, then A OR B = {1,2,3,4,5,6,7,8}.

• Intersection (AND) Event: The event A AND B includes only outcomes in both A and B. Example: If A = {1,2,3,4,5} and B = {4,5,6,7,8}, then A AND B = {4,5}.

Complement Rule

The complement of an event consists of all outcomes not in the event. The probability of the complement is easily calculated.

• The complement of event A is denoted Ac or A'.

• P(A) + P(A') = 1

• Example: If S = {1,2,3,4,5,6} and A = {1,2,3,4}, then A' = {5,6}. $P(A) = \\frac{4}{6}$, $P(A') = \\frac{2}{6}$