3.1 Terminology

Probability is the backbone of statistical analysis, providing a framework to measure uncertainty and make predictions. This section introduces key concepts like experiments, outcomes, and events, laying the groundwork for understanding how probabilities are calculated and interpreted.

Diving deeper, we explore the difference between "and" and "or" events, conditional probability, and independence. These concepts are crucial for solving complex probability problems and form the basis for more advanced statistical techniques used in data analysis and decision-making.

Probability Terminology and Concepts

Key probability terminology

• Experiment involves a process or action that generates well-defined outcomes (rolling a die, flipping a coin, drawing a card from a deck)
• Outcome represents a single result of an experiment, also known as a sample point (rolling a 3 on a die, flipping a head on a coin, drawing an ace from a deck)
• Sample space encompasses the set of all possible outcomes of an experiment, denoted by $S$ ($S = {1, 2, 3, 4, 5, 6}$ for rolling a die, $S = {H, T}$ for flipping a coin)
• Event consists of a subset of the sample space, a collection of one or more outcomes (rolling an even number on a die, flipping at least one head in two coin flips)
• Random variable is a function that assigns a numerical value to each outcome in the sample space

Probability calculation for equal outcomes

• Probability of an event measures the likelihood of an event occurring, denoted by $P(A)$ for event $A$
  • Formula: $P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$
• Equally likely outcomes occur when each outcome in the sample space has the same probability of occurring (each side of a fair die has a probability of $\frac{1}{6}$)
• Calculating probabilities involves:
  1. Identifying the sample space and the total number of possible outcomes
  2. Determining the number of favorable outcomes for the event of interest
  3. Dividing the number of favorable outcomes by the total number of possible outcomes

"And" vs "or" events

• "AND" events (Intersection), denoted by $A \cap B$, occur when both event $A$ and event $B$ happen simultaneously (rolling a 3 AND flipping a head)
  • Probability of "AND" events: $P(A \cap B) = P(A) \times P(B)$ if $A$ and $B$ are independent
• "OR" events (Union), denoted by $A \cup B$, occur when either event $A$ or event $B$ or both happen (rolling a 3 OR flipping a head)
  • Probability of "OR" events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$
• Mutually exclusive events cannot occur simultaneously
  • If $A$ and $B$ are mutually exclusive, $P(A \cap B) = 0$
  • For mutually exclusive events, $P(A \cup B) = P(A) + P(B)$

Advanced Probability Concepts

• Conditional probability is the probability of an event occurring given that another event has already occurred
• Independent events are events where the occurrence of one does not affect the probability of the other
• Law of large numbers states that as the number of trials increases, the sample mean approaches the expected value
• Expected value is the average outcome of an experiment if it is repeated many times