Basic Concepts of Probability: Theoretical and Empirical Approaches

Basic Concepts of Probability

Introduction to Probability

Probability is a fundamental concept in statistics that quantifies how likely an event is to occur. It is expressed as a number between 0 and 1, where 0 means the event cannot happen and 1 means the event is certain to happen. The probability of an event is denoted as P(event).

• Probability Formula: The general formula for probability is: $P(\text{event}) = \frac{\text{number of ways the event can occur}}{\text{total number of possible outcomes}}$

Theoretical vs. Empirical Probability

Theoretical Probability

Theoretical probability is determined by reasoning or calculation, without conducting any actual experiment. It is based on what should happen in an ideal situation.

• Definition: Theoretical probability is calculated as: $P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$

• Example: When rolling a six-sided die, the probability of rolling a 2 is: $P(2) = \frac{1}{6}$ Explanation: There is only one outcome (rolling a 2) out of six possible outcomes (1, 2, 3, 4, 5, 6).

Empirical (Experimental) Probability

Empirical probability is based on actual experiments or observations. It is calculated using data collected from performing the experiment multiple times.

• Definition: Empirical probability is calculated as: $P(\text{event}) = \frac{\text{number of times the event occurred}}{\text{total number of trials}}$

• Example: If you roll a die 10 times and get a 2 three times, the empirical probability of rolling a 2 is: $P(2) = \frac{3}{10}$

Sample Space

Definition and Representation

The sample space is the set of all possible outcomes of an experiment. It is often represented using curly braces { }.

• Example: The sample space for flipping a coin is {H, T}, where H = heads and T = tails.

Practice Problems

Probability from Tabular Data

Probability can be calculated using data from tables that show the frequency of outcomes.

Group	Wearing Jeans	Not Wearing Jeans
1	68	27
2	17	17

• Example: Probability that a person randomly selected from Group 1 is wearing jeans: $P(\text{wearing jeans}) = \frac{68}{68 + 27} = \frac{68}{95}$

Probability with Coins

• Example: If you have 3 quarters, 4 nickels, 6 dimes, and 7 pennies, the probability of picking a quarter at random is: $P(\text{quarter}) = \frac{3}{3 + 4 + 6 + 7} = \frac{3}{20}$

Summary Table: Theoretical vs. Empirical Probability

Type	Definition	Formula	Example
Theoretical	Based on reasoning; what should happen	$P(\text{event}) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}$	Probability of rolling a 2 on a die: $\frac{1}{6}$
Empirical	Based on experiment; what did happen	$P(\text{event}) = \frac{\text{number of times event occurred}}{\text{total number of trials}}$	Probability of rolling a 2 three times in 10 rolls: $\frac{3}{10}$

Additional info: These notes cover foundational probability concepts relevant to Chapter 5: Probability in Our Daily Lives, including definitions, formulas, and practical examples for both theoretical and empirical probability.