Question

Under which of the following circumstances is the pair \(A\), \(B\) of events in sample space \(\Omega\) an independent pair? Explain your answer. (a) \(A\) and \(B\) are disjoint, \(P(A)>0,\) and \(P(B)>0\) (b) \(P(A)=0\) and \(P(B)>0\) (c) \(P(A)=P(B)=0\)

Short answer

Events are independent in cases (b) and (c).

Step-by-step solution

Understanding Independence

Two events, \(A\) and \(B\), in sample space \( \Omega \) are independent if the probability of their intersection is equal to the product of their probabilities: \( P(A \cap B) = P(A)P(B) \). Let's analyze each given condition to see if they satisfy this property.

Evaluating Case (a)

In case (a), \( A \) and \( B \) are disjoint with \( P(A) > 0 \) and \( P(B) > 0 \). Since \( A \) and \( B \) are disjoint, \( A \cap B = \emptyset \), meaning \( P(A \cap B) = 0 \). However, their independence would require \( P(A \cap B) = P(A)P(B) \), and since \( P(A), P(B) > 0 \), \( P(A)P(B) > 0 \). Therefore, \( 0 eq P(A)P(B) \), so \( A \) and \( B \) are not independent in this case.

Evaluating Case (b)

In case (b), \( P(A) = 0 \) and \( P(B) > 0 \). Here, \( P(A \cap B) = 0 \) since the intersection cannot have a positive probability if \( A \) has a probability of zero. Since \( P(A) = 0 \), we have \( P(A)P(B) = 0 \cdot P(B) = 0 \). This satisfies \( P(A \cap B) = P(A)P(B) \), so \( A \) and \( B \) are independent.

Evaluating Case (c)

In case (c), both \( P(A) = 0 \) and \( P(B) = 0 \). Here, \( P(A \cap B) = 0 \) because neither \( A \) nor \( B \) can occur. The product \( P(A)P(B) = 0 \cdot 0 = 0 \). Thus, \( P(A \cap B) = P(A)P(B) \), indicating that \( A \) and \( B \) are independent.

Key concepts

Sample Space

In probability theory, a **sample space** is the set of all possible outcomes of an experiment. Consider rolling a standard six-sided die. The sample space, in this case, would be \{1, 2, 3, 4, 5, 6\}, as each number represents a potential outcome. A sample space can be finite or infinite, depending on the nature of the experiment. It serves as the foundational basis upon which events and probabilities are defined.

In the context of events, it is crucial to understand how specific subsets of the sample space determine the occurrence of these events. Each event is essentially a collection of outcomes from the sample space. An event that consists of all the possible outcomes is called a certain event, while an event that includes none of the outcomes is known as a null event or impossible event.

• Sample space \( \Omega \) encapsulates all possible outcomes.
• Events are subsets within this sample space.
• Understanding the sample space helps in framing probabilities for different events.

Intersection of Events

The **intersection of events** refers to the scenario where two or more events occur simultaneously. Mathematically, the intersection of two events, \( A \) and \( B \), is denoted as \( A \cap B \). This intersection includes all the outcomes that are common to both events.

For instance, when flipping two coins, let \( A \) represent the event of getting a head on the first coin, and \( B \) represent getting a head on the second coin. The intersection \( A \cap B \) would include the outcome \('HH'\), where both coins show heads.

• The intersection helps determine the probability of both events occurring together.
• Indicates shared outcomes between events.
• Critical for calculating probabilities in compound events.

Probability of Events

The **probability of events** measures the likelihood of an event occurring, calculated within a given sample space. Its value ranges from 0 to 1, where 0 indicates impossibility and 1 denotes certainty. The formula to determine the probability of an event \( A \) is \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}} \).

For simple events, such as rolling a die, calculating probability is straightforward. However, complexities arise when dealing with compound events, such as intersections or unions of multiple events. Understanding these concepts helps in accurately evaluating various event combinations.

• Represents the chance of an event occurring.
• Can be applied to simple or complex event sets.
• Essential for analyzing the behavior of random processes.