Problem 25 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

An experiment consists of selecting a card from a standard deck of playing cards and noting whether it is black \((B)\) or red \((R)\). a. Describe an appropriate sample space for this experiment. b. What are the events of this experiment?

Step-by-Step Solution

Answer

The sample space for this experiment, where a card is selected from a standard deck and its color is noted, is given by \(S = \{B, R\}\), where \(B\) represents a black card and \(R\) represents a red card. In this context, the events of the experiment are defined as the drawing of a black card, which can be designated as event \(A\) and represented as \(A = \{B\}\), and the drawing of a red card, which can be designated as event \(B\) and represented as \(B = \{R\}\).

Step 1: Identifying the sample space

In a standard deck of 52 playing cards, there are 26 black cards (spades and clubs) and 26 red cards (hearts and diamonds). The required sample space consists of all the possible outcomes of drawing a single card from the deck. Since there are only two colors, black and red, the sample space for this experiment can be described as: \[S = \{B, R\}\]

Step 2: Identifying the events

An event in probability theory refers to a specific outcome or a set of outcomes that we are interested in. In this experiment, we are interested in the color of the cards. There are two possible events: drawing a black card and drawing a red card. To represent these events in terms of our sample space, we can define event \(A\) as drawing a black card and event \(B\) as drawing a red card: Event \(A\): Drawing a black card \(\Rightarrow A = \{B\}\) Event \(B\): Drawing a red card \(\Rightarrow B = \{R\}\) In summary, the sample space for this experiment is given by \(S = \{B, R\}\), and the two events of interest are the drawing of a black card (event \(A\)) and the drawing of a red card (event \(B\)).

Key Concepts

Understanding Sample Space

When delving into probability theory, one of the foundational concepts we encounter is the sample space. This term refers to the complete set of all possible outcomes of a random experiment. For instance, when we're talking about flipping a coin, the sample space consists of two elements - heads or tails. The concept becomes even more interesting when we deal with more complex scenarios, such as drawing cards from a deck.

Let's relate this back to our textbook exercise: when selecting a card from a standard deck, the sample space is simplified due to the nature of the experiment, focusing only on color. Thus, the sample space, denoted by the symbol \(S\), includes just two outcomes: black, \(B\), and red, \(R\). It's important to remember that a well-defined sample space should be mutually exclusive (each outcome is distinct) and collectively exhaustive (no other outcomes exist).

Sample Space: \(S = \{B, R\}\)
It is crucial for students to grasp that establishing the sample space is a critical step before analyzing probability events as it frames the context for further calculations and observations.

Demystifying Probability Events

Next, we need to understand probability events. In our context, an event is a set of outcomes from the sample space that we are interested in evaluating. We might seek the likelihood of these outcomes individually or collectively. A key factor in probability is determining how many ways an event can occur relative to all possible outcomes.

In the case where we are drawing cards from a deck, our exercise is narrowed down to color, hence there are only two events as outlined previously. Event \(A\) is drawing a black card, whereas event \(B\) is drawing a red card. In probability notation, we represent these events as subsets of the sample space:

• Event \(A\) (Black Card): \(A = \{B\}\)
• Event \(B\) (Red Card): \(B = \{R\}\)

Understanding events helps you conceptualize what you're actually computing when you work out probabilities. It’s the bread and butter of probability theory, determining 'what are the chances', for the outcomes defined within the sample space.

Navigating a Standard Deck of Cards

Probably one of the most popular contexts for probability problems is a standard deck of cards. When we discuss a 'standard deck', we're referring to 52 playing cards with 4 suits: spades (black), clubs (black), diamonds (red), and hearts (red). With each suit having 13 cards, from Aces to Kings.

Each suit in a deck is split into two colors: 26 cards are black, and 26 are red. This is essential information when analyzing probability events related to card colors. Since we're focused on color, it simplifies our perspective, but remember that many more complex probability exercises may require consideration of suits, ranks, or combinations thereof.

Understanding the makeup of a deck is vital for probability calculations. Knowing there are equal numbers of black and red cards can influence the likelihood of an event occurring. For example, the probability of drawing a black or red card from a fully shuffled deck is equal, at 50%, since both outcomes have an equal number of favorable cases within the total number of possibilities.