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. $$ E^{\prime} \cap F^{\prime} $$

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Key Concepts

Sample Space

The sample space is the set of all possible outcomes of a probabilistic experiment. In problems involving dice, it includes all ordered pairs representing the results from each die when the dice are rolled. Understanding the sample space allows one to define events as subsets of the possible outcomes.

Complement of an Event

The complement of an event consists of all outcomes in the sample space that are not part of the event itself. It is used to represent scenarios where a certain condition is not satisfied, and it plays a key role in calculations involving probability and set operations.

Intersection of Events

The intersection of two events is the set of outcomes that are common to both events. This key concept helps in determining situations where multiple conditions hold simultaneously, a frequent requirement in probability and set theory problems.

De Morgan's Laws

De Morgan's Laws provide a fundamental relationship between the union and intersection of sets through complementation. In the given context, they offer insights into transforming expressions involving complements of events, such as showing that the intersection of two complements is equivalent to the complement of the union of the events.

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