Problem 16 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(S\) be any sample space and let E, \(\boldsymbol{F}\), and \(\boldsymbol{G}\) be any three events associated with the experiment. Describe the events using the symbols \(\cup, \cap\), and . The event that both \(E\) and \(F\) occur

Step-by-Step Solution

Answer

The event that both E and F occur can be represented as: \(E \cap F\).

Step 1: Identify the events and symbols

We are given the events E and F and are asked to represent the event where both E and F occur using the symbols ∪, ∩, and '. In this case, we will use the intersection symbol (∩) as it represents the elements that are common in both events.

Step 2: Represent the event

Since the intersection symbol (∩) is used for the elements that are common in both events, we can represent the event that both E and F occur as: \(E \cap F\).

Step 3: Interpret the result

The event \(E \cap F\) represents the event that both E and F occur, which means it includes all the outcomes that are present in both event E and event F.

Key Concepts

Sample Space

In probability theory, the concept of a sample space is fundamental. It's essentially the set of all possible outcomes of a particular experiment or random trial. Picture it as the complete list of everything that can happen when you perform a random experiment.

• If you toss a coin, the sample space is {Heads, Tails}.
• For rolling a six-sided die, the sample space includes all six sides: {1, 2, 3, 4, 5, 6}.

Sample spaces can either be finite or infinite, depending on the experiment at hand.
When you're dealing with cards, dice, coins, or other similar items, the sample space is usually finite.
However, if you consider an experiment like picking a real number between 0 and 1, the sample space becomes infinite.
Understanding what your sample space looks like is crucial because it lays down the foundation for computing probabilities and understanding events.

Intersection of Events

The intersection of events refers to the scenario where multiple events occur simultaneously. In probability theory, to represent this, we use the symbol ∩. This symbol implies the shared or common outcomes between two or more events. Consider two events, E and F. The intersection of E and F, denoted as \(E \cap F\), contains all the outcomes that are present in both events. Here's a simple example to illustrate this concept:
Suppose you have a standard deck of cards.
Let event E be drawing a heart, and event F be drawing a face card. The intersection, \(E \cap F\), would be the outcomes where a card drawn is both a heart and a face card.
This would include the Jack, Queen, and King of hearts.

• In graphical terms, if you imagine each event as a circle, \(E \cap F\) would be the overlapping region of these circles.
• The intersection is critical when determining probabilities involving dependent events.

Events in Probability

In probability theory, an event is essentially a subset of the sample space. It's a specific set of outcomes that we are interested in when considering an experiment. Understanding events is crucial, as they form the basis for defining probabilities. Here's how we can classify events in probability:

• Simple Events: These involve just a single outcome from the sample space. For example, rolling a 3 on a six-sided die.
• Compound Events: These consist of two or more simple events. An example would be rolling an even number on a die.

Events can also be classified as independent or dependent, depending on how they relate to each other. Independent events have no impact on each other's outcomes, whereas dependent events do.
Calculating probabilities often involves determining the number of favorable outcomes (our event of interest) divided by the total number of outcomes in the sample space. This ratio is central to how probabilities are expressed and understood.
Understanding these concepts helps in solving complex probability problems and determining the likelihood of various outcomes.