1.2 Sample spaces, events, and outcomes

Random experiments form the backbone of probability theory. Sample spaces represent all possible outcomes, while events are subsets of these spaces. Understanding these concepts is crucial for analyzing and predicting random phenomena.

Discrete sample spaces have countable outcomes, while continuous ones have infinite possibilities. Events can be mutually exclusive (no overlap) or exhaustive (cover all outcomes). These ideas lay the foundation for calculating probabilities and making informed decisions.

Sample Spaces and Events

Sample spaces in random experiments

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• Represents the set of all possible outcomes in a random experiment
  • Uses the notation $S$ to denote the sample space
  • Rolling a six-sided die has a sample space of $S = \{1, 2, 3, 4, 5, 6\}$
  • Tossing a coin twice results in a sample space of $S = \{HH, HT, TH, TT\}$
• Individual outcomes within the sample space are known as sample points
• Set notation or tree diagrams can be used to represent the sample space

Discrete vs continuous sample spaces

• Discrete sample spaces have a finite or countably infinite number of outcomes
  • Rolling a die, tossing a coin, and counting defective items in a batch are discrete
• Continuous sample spaces have an uncountably infinite number of outcomes
  • Typically represented by intervals of real numbers
  • Temperature, weight, and time are examples of continuous sample spaces

Events as sample space subsets

• Subsets of the sample space are called events
  • Denoted by capital letters like $A$ or $B$
  • Rolling a die and getting an even number is an event $A = \{2, 4, 6\}$
  • Tossing a coin twice and getting at least one head is an event $B = \{HH, HT, TH\}$
• The empty set $\emptyset$ and the sample space $S$ are also considered events
• The complement of an event $A$, denoted by $A^c$, contains all outcomes in the sample space that are not in $A$

Mutually exclusive and exhaustive events

• Mutually exclusive events $A$ and $B$ have no outcomes in common
  • Formally expressed as $A \cap B = \emptyset$
  • Rolling a die and getting an even number ($A$) or an odd number ($B$) are mutually exclusive
• Exhaustive events cover the entire sample space when combined
  • If $A_1, A_2, \ldots, A_n$ are exhaustive events, then $A_1 \cup A_2 \cup \ldots \cup A_n = S$
  • Rolling a die and getting a number less than 4 ($A = \{1, 2, 3\}$) or greater than or equal to 4 ($B = \{4, 5, 6\}$) are exhaustive events since $A \cup B = \{1, 2, 3, 4, 5, 6\} = S$