Let S be the set of outcomes when two distinguishable dice...

Let $S$ be the set of outcomes when two distinguishable dice are rolled, let $E$ be the subset of outcomes in which at least one die shows an even number, and let $F$ be the subset of outcomes in which at least one die shows an odd number. List the elements in each subset given. $$ (E \cap F)^{\prime} $$

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Key Concepts

Sample Space

The sample space in probability is the set of all possible outcomes of an experiment. In problems involving dice, it typically consists of all ordered pairs representing the result on each die. Understanding the sample space is crucial for analyzing probabilities because it provides the framework within which events are defined and evaluated.

Event

An event is any subset of the sample space, representing outcomes that share a specific property or satisfy certain conditions. In this context, events are described by conditions such as having at least one even number or at least one odd number. Recognizing and defining events accurately allows for the application of probability theorems and set operations.

Set Operations (Intersection and Complement)

Set operations are tools from set theory used to combine or modify events. The intersection of two events represents outcomes common to both, while the complement of an event consists of all outcomes in the sample space that are not in the event. These operations are essential for solving problems that involve conditions such as 'at least one even and at least one odd' and finding outcomes that do not meet a combined condition.

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