Question

A sample space consists of \(S=\left\\{E_{1}, E_{2}\right.\), \(\left.E_{3}, E_{4}\right\\} .\) List the simple events in "both \(A\) and \(B\)," " \(A\) or \(B\) or both," and "not \(B\) " for the events given in Exercises \(13-15 .\) $$A=\left\\{E_{1}\right\\} \text { and } B=\left\\{E_{1}, E_{3}, E_{4}\right\\}$$

Short answer

Answer: (1) The simple events in both A and B are $\{E_1\}$. (2) The simple events in A or B or both are $\{E_1, E_3, E_4\}$. (3) The simple events in not B are $\{E_2\}$.

Step-by-step solution

Find both A and B (Intersection)

To find the simple events that are in both A and B, we have to find the intersection of the two sets. The intersection of sets A and B is defined as the set of elements that are present in both sets. Mathematically, this can be represented as \(A \cap B\). In our case, given the events: $$A = \{E_1\}, B = \{E_1, E_3, E_4\}$$ The intersection of events A and B can be found as: $$A \cap B = \{E_1\}$$

Find A or B or both (Union)

To find the simple events that are in either A, B or both, we have to find the union of the two sets. The union of sets A and B is defined as the set of elements that are present in either of the sets or both. Mathematically, this can be represented as \(A \cup B\). In our case, given the events: $$A = \{E_1\}, B = \{E_1, E_3, E_4\}$$ The union of events A and B can be found as: $$A \cup B = \{E_1, E_3, E_4\}$$

Find not B (Complement)

To find the simple events not in B (event not B), we have to find the complement of set B. The complement of set B is defined as the set of elements that are not present in set B but are in the sample space S. Mathematically, this can be represented as \(\overline{B}\) or \(S - B\). In our case, given the sample space and event B: $$S=\{E_1, E_2, E_3, E_4\}, B=\{E_1, E_3, E_4\}$$ The complement of event B can be found as: $$\overline{B} = S - B = \{E_2\}$$

Key concepts

Sample Space

In probability and statistics, the sample space is the foundation upon which probability events are built. It represents all the possible outcomes of an experiment or random trial. For instance, if we toss a fair coin, the sample space consists of two outcomes, heads (\text{H}) or tails (\text{T}).

When dealing with more complex experiments, the sample space can be a large set consisting of multiple elements. In the problem given, the sample space is denoted by:
\( S = \{E_1, E_2, E_3, E_4\} \),
where \(E_1, E_2, E_3,\text{ and }E_4\) represent the different simple events that could occur. Defining the sample space accurately is crucial to understanding and calculating probabilities effectively, since all probabilities are related to this initial set of possibilities.

Set Theory in Probability

Set theory in probability is used to describe how different events relate to each other within the sample space. Probabilistic events can be thought of like sets, with elements of those sets representing outcomes that satisfy specific conditions. For example, if we have an event that an ace is drawn from a deck of cards, we can represent it by the set of all aces in the deck.

Several operations from set theory are employed in probability:

• Union (\text{denoted by} \(\cup\)) represents the combination of all outcomes in either set or both.
• Intersection (\text{denoted by} \(\cap\)) represents outcomes that are common to both sets.
• Complement of a set includes all outcomes in the sample space that are not in the given set.

By applying these operations, probability theory provides a structured way to calculate the likelihood of various events.

Union and Intersection of Sets

When we're working with multiple events in probability, we can combine them using the concepts of union and intersection.

The union of two sets contains all the elements that are in either set, or in both. It's like telling a story where all characters from two books are combined. Mathematically, for two sets \(A\) and \(B\), it is denoted as \(A \cup B\). In our example,
\(A \cup B = \{E_1, E_3, E_4\}\),
includes all elements from both events \(A\) and \(B\).

On the other hand, the intersection of two sets contains only the elements that are in both sets. It's like a crossover where only the characters that appear in both books are featured. Mathematically, it is denoted as \(A \cap B\). In our example,
\(A \cap B = \{E_1\}\),
which shows that \(E_1\) is the only outcome present in both events \(A\) and \(B\).

These operations help us to determine the probability of combined events, whether we're interested in the likelihood of either event occurring or both events occurring at the same time.