Probability: Basic Concepts

Introduction to Probability

Probability is a foundational concept in statistics, quantifying the likelihood of events occurring in a random experiment. Probability values range from 0 (impossible event) to 1 (certain event), and are used to interpret and predict outcomes in various contexts, such as games of chance, scientific studies, and everyday decision-making.

• Event: Any collection of results or outcomes of a procedure.

• Simple Event: An outcome that cannot be further broken down into simpler components.

• Sample Space: The set of all possible simple events for a procedure.

Examples of Simple Events and Sample Spaces

• Single Birth: Sample space = {b, g} (b = boy, g = girl).

• Three Births: Sample space = {bbb, bbg, bgb, bgg, gbb, gbg, ggb, ggg}.

• Rolling a Die: The outcome of rolling a 5 is a simple event; the outcome of rolling an even number is not a simple event, as it can be broken down further.

Three Common Approaches to Finding Probability

Probability Notation and Range

• P(A): Probability of event A occurring.

• Probability values satisfy .

1. Relative Frequency Approximation

This approach estimates probability based on the outcomes of repeated trials of an experiment.

• Formula:

• Example: If 16 out of 39,000,000 airline flights crashed,

2. Classical Approach (Equally Likely Outcomes)

Used when all simple events are equally likely.

• Formula:

• Caution: Only use when outcomes are equally likely.

3. Subjective Probability

Probability is estimated based on knowledge, experience, or intuition about the situation.

• Example: Estimating the chance of rain based on weather patterns.

Simulations

When none of the above approaches are feasible, simulations can be used to model a procedure and estimate probabilities by mimicking real-life processes.

Rounding Probabilities

• Express probabilities as exact fractions or decimals, or round to three significant digits for clarity.

• Example: or

Worked Examples

Example: Probability from Survey Data

• 366 adults answered "yes" to seeing a ghost; 1637 answered "no"; total responses = 2003.

• Probability:

• Interpretation: There is an 18.3% chance a randomly selected adult reports seeing a ghost.

Significance in Probability

Identifying Significant Results

• Significantly High: x successes among n trials is significantly high if .

• Significantly Low: x successes among n trials is significantly low if .

• The threshold 0.05 is commonly used but not absolute.

Complementary Events

Definition and Notation

• Complement of A (denoted ): All outcomes where event A does not occur.

• Probability of Complement:

Probability Review

• Probability is always between 0 and 1 (inclusive).

• Impossible event:

• Certain event:

• Notation: for event A, for the complement of A.

Odds

Actual Odds Against and In Favor

• Actual Odds Against A: , usually expressed as a:b.

• Actual Odds In Favor of A: , the reciprocal of odds against.

• If odds against are a:b, odds in favor are b:a.

Example: Roulette Odds

• Probability of winning on 13:

• Probability of not 13:

• Actual odds against 13:

• Casino payoff odds: 35:1 (less than actual odds, favoring the casino)

• If payoff odds matched actual odds (37:1), a $5 bet would yield $185 profit.

Summary Table: Probability Approaches