Question

Suppose \(A\) and \(B\) are disjoint events in a sample space \(\Omega\). Is it possible that \(A\) and \(B\) could be independent? Explain your answer.

Short answer

Disjoint events cannot be independent unless one or both have zero probability.

Step-by-step solution

Understanding Disjoint Events

Disjoint events, also known as mutually exclusive events, are events that cannot occur at the same time. This means that if event \(A\) occurs, event \(B\) does not, and vice versa. In probability terms, this is expressed as \( P(A \cap B) = 0 \).

Definition of Independent Events

Two events are independent if the occurrence of one does not affect the occurrence of the other. For events \(A\) and \(B\), they are independent if \( P(A \cap B) = P(A) \times P(B) \).

Comparing Definitions

Disjoint events have \( P(A \cap B) = 0 \), which is incompatible with the condition for independence \( P(A \cap B) = P(A) \times P(B) \) unless \( P(A) = 0 \) or \( P(B) = 0 \). Therefore, if events are actually occurring (having non-zero probabilities), being disjoint means they cannot be independent.

Final Analysis

If either \( P(A) = 0 \) or \( P(B) = 0 \), then the condition for independence (\( P(A \cap B) = P(A) \times P(B) \)) holds true trivially because \(0 \cdot P(B) = 0\) or \(P(A) \cdot 0 = 0\). However, in practical scenarios with non-zero probabilities, disjoint events cannot be independent.

Key concepts

Disjoint Events

Disjoint events, also referred to as mutually exclusive events, are a key concept in probability theory. These are events that cannot happen simultaneously. Simply put, if event \(A\) occurs, event \(B\) cannot occur at the same time, and vice versa. This is mathematically represented by the formula \( P(A \cap B) = 0 \).

In this context, \( A \cap B \) indicates the overlapping section of \(A \) and \(B\), which is the part where both events occur concurrently. Since disjoint events do not overlap, this intersection is \(0\). Thus, for disjoint events, not only is their occurrence exclusive, but the probability of both happening together is zero.

For instance, when flipping a coin, the outcomes "heads" and "tails" are disjoint. Naturally, both cannot appear at once.

Independent Events

In probability theory, independent events are events whose occurrence does not influence each other. If two events, \(A\) and \(B\), are independent, the probability of both occurring is the product of their individual probabilities.

This relationship is defined as \( P(A \cap B) = P(A) \times P(B) \). Here, \( P(A \cap B) \) refers to the likelihood of both \(A\) and \(B\) occurring. When two events are independent, knowing the outcome of one provides no information about the other. This lack of influence is a crucial feature of independence in probability.

For example, if you roll a die and flip a coin, the result of the coin flip (heads or tails) does not impact the number rolled on the die. Therefore, the events are independent.

Mutually Exclusive Events

While disjoint and mutually exclusive events are often terms used interchangeably, it is useful to highlight their shared characteristics. Mutually exclusive events, just like disjoint events, cannot happen at the same time.

These events have a probability formula used to show that \( P(A \cap B) = 0 \), reflecting their incompatibility with occurring together. If one event happens, the other simply cannot. This makes the occurrence of mutually exclusive events strictly one or the other, with no overlap or simultaneous occurrence.

A classic illustration is with dice: rolling a \(4\) and a \(5\) on a single die roll is mutually exclusive. It's impossible to land on both numbers at once, showing the mutual exclusivity of each event's outcome.