Problem 25 - Applied Statistics and Probability for Engineers

Question

In control replication, cells are replicated over a period of two days. Not until mitosis is completed can freshly synthesized DNA be replicated again. Two control mechanisms have been identified-one positive and one negative. Suppose that a replication is observed in threc cells. Let \(A\) denote the event that all cells are identified as positive, and let \(B\) denote the event that all cells are negative. Describe the sample space graphically and display each of the following events: (a) A (b) \(B\) (c) \(A \cap B\) (d) \(A \cup B\)

Step-by-Step Solution

Answer

(a) \(A = \{ (+++) \}\); (b) \(B = \{ (---) \}\); (c) \(A \cap B = \emptyset\); (d) \(A \cup B = \{ (+++), (---) \}\).

Step 1: Understand the Sample Space

First, we need to establish the sample space for the exercise. Since each cell can be either 'positive' (+) or 'negative' (-), for three cells, the sample space consists of all combinations of these outcomes. Represent each cell's state using symbols "+" or "-". The complete sample space (all possible combinations) is: 1. (+++), 2. (++-), 3. (+-+), 4. (-++), 5. (+--), 6. (-+-), 7. (--+), 8. (---). These represent all possible outcomes for three cells.

Step 2: Define Event A (All Cells Positive)

Event \(A\), where all cells are identified as positive, means each cell in the set of three is "+". In our sample space, the combination corresponding to this event is (+++). Thus, \(A = \{(+++)\}\).

Step 3: Define Event B (All Cells Negative)

Event \(B\), where all cells are identified as negative, means each cell in the set of three is "-". In our sample space, the combination corresponding to this event is (---). Therefore, \(B = \{(---)\}\).

Step 4: Identify the Intersection of A and B

The intersection of events \(A \cap B\) refers to the outcomes that are common to both event \(A\) and event \(B\). Since event \(A\) represents all positives (+++) and event \(B\) represents all negatives (---), there are no common elements between the two. Thus, \(A \cap B = \emptyset\) (the empty set).

Step 5: Determine the Union of A and B

The union of events \(A \cup B\) includes any outcomes that are in \(A\) or \(B\) or in both. From the sample space, \(A\) is (+++) and \(B\) is (---). The union of these events is \(A \cup B = \{ (+++), (---) \}\). This means the outcomes are fulfilled when all cells are either entirely positive or entirely negative.

Key Concepts

Sample Space

In probability theory, the sample space is the foundation for understanding events as it contains all the possible outcomes of an experiment. Consider it as the universe of our problem. In the exercise, we look at three cells, each of which can be positive or negative. So, the sample space consists of every possible combination of these states.
To visualize, we list each potential outcome:

• (+++), where all are positive.
• (++-), where the first two are positive and the last is negative.
• (+-+), where the first and third are positive and the second is negative.
• (-++), where the last two are positive and the first is negative.
• (+--), where the first is positive and the last two are negative.
• (-+-), where the first and third are negative, and the second is positive.
• (--+), where the first two are negative and the third is positive.
• (---), where all are negative.

This array of combinations forms our sample space – every potential arrangement of positive and negative states for the three cells.

Events

An event in probability is a specific set of outcomes from the sample space. Think of it as a scenario with particular rules.

• Event A: All cells are positive, represented by (+++).
• Event B: All cells are negative, shown as (---).

Unlike the entire sample space, which includes all possible outcomes, events are more specific. They allow us to focus on scenarios with certain desired characteristics.
For instance, in this exercise, event A (all positive) and event B (all negative) have specific outcomes, and we use these as building blocks to explore more complex event relationships.

Intersection and Union

Intersections and unions are crucial concepts in probability that help us analyze relationships between events.
When we talk about the **intersection** of two events, denoted as \(A \cap B\), we're looking for outcomes that both events share. It refers to scenarios where both characteristics can be true simultaneously. In this exercise, event A (all positive) and event B (all negative) have no overlapping outcomes, so their intersection is empty, \(A \cap B = \emptyset\).
On the other hand, the **union** of two events, represented as \(A \cup B\), means we include any outcome that is true for either event A or event B or for both. In our exercise, the union of A and B includes the outcomes (+++) from A and (---) from B. Thus, \(A \cup B = \{(+++) , (---)\}\). Finding these allows us to predict the possibility of either of these scenarios happening in the experiment.

Set Theory

Set theory forms the language of probability and statistics, providing a framework for organizing and analyzing data. In this context, it's about dealing with collections of outcomes, or sets, like the sample space or events A and B. Each outcome or combination is an element of these sets.
We represent and manipulate these sets using specific operations:

• The **empty set** \(\emptyset\) denotes a set with no elements. For example, the intersection \(A \cap B\) is an empty set because there are no common outcomes between (+++) and (---).
• **Unions** and **intersections** are operations applied to sets to explore combinations and shared components, as we discussed.

This mathematical structure helps us navigate through probability problems by mapping complex ideas into more digestible forms, making it easier to draw conclusions about our data and predictions.