Question

(a) Give an example that shows three pairwise independent events need not be an independent set of events. (b) Give an example that shows three events can be independent without having the corresponding pairs of events be independent.

Short answer

(a) A, B, and C are pairwise but not jointly independent if they do not satisfy joint probability conditions. (b) Events can be jointly independent without pairwise independence by violating pairwise conditions while maintaining the joint condition.

Step-by-step solution

Understanding Pairwise Independence

Pairwise independence of three events \( A, B, \) and \( C \) means that every pair of events is independent. This can be demonstrated with probabilities such that \( P(A \cap B) = P(A)P(B) \), \( P(B \cap C) = P(B)P(C) \), and \( P(A \cap C) = P(A)P(C) \). However, this does not imply joint independence, which would require \( P(A \cap B \cap C) = P(A)P(B)P(C) \).

Example for Part (a): Pairwise but Not Joint Independence

Consider three events in a probability space: \( A = \{1, 2\}\), \( B = \{1, 3\}\), and \( C = \{2, 3\}\), defined over a space \( \Omega = \{1, 2, 3, 4\} \) with equal probability \( 0.25 \) for each outcome. Here, \( P(A) = P(B) = P(C) = 0.5 \). Calculating each intersection: \( P(A \cap B) = 0.25, P(B \cap C) = 0.25, P(A \cap C) = 0.25 \), which equal \( P(A)P(B), P(B)P(C), \) and \( P(A)P(C) \). However, \( P(A \cap B \cap C) = 0 \) which is not equal to \( P(A)P(B)P(C) = 0.125 \). So these events are pairwise but not jointly independent.

Understanding Joint Independence

Joint independence of three events \( A, B, \) and \( C \) requires that not only every pair of events must be independent, but also \( P(A \cap B \cap C) = P(A)P(B)P(C) \).

Example for Part (b): Joint Independence without Pairwise Independence

For this case, consider a probability space from four fair coin tosses: Let \( A \) be the event that the first coin is Heads, \( B \) that the second is Heads, and \( C \) that the total number of Heads is even. Analyze the following outcomes: Heads-Heads-Heads-Heads, Heads-Heads-Tails-Tails, Tails-Tails-Heads-Tails, Tails-Heads-Heads-Heads. Here, outcomes can be chosen such that \( P(A \cap B) = P(A)P(B) \) and \( P(A \cap C) = P(A)P(C) \) do not hold, thus making them individually independent but not pairwise.

Key concepts

Pairwise Independence

Pairwise independence is a fascinating concept that appears in probability theory, notably when analyzing independent events. When we say that events \(A, B,\) and \(C\) are pairwise independent, it implies that each pair from these events is independent of each other. The mathematical condition for this is:

• \(P(A \cap B) = P(A)P(B)\)
• \(P(B \cap C) = P(B)P(C)\)
• \(P(A \cap C) = P(A)P(C)\)

However, a surprising twist is that while each pair might behave independently, this does not guarantee that all three events will be independent when considered together. Add the requirement for joint independence, and the set of conditions becomes more stringent. In the provided example, the events \(A = \{1, 2\}\), \(B = \{1, 3\}\), and \(C = \{2, 3\}\) are pairwise independent when calculated, but they are not jointly independent because \(P(A \cap B \cap C) = 0\), which does not satisfy \(P(A)P(B)P(C) = 0.125\). Keep this interesting caveat in mind: pairwise independence doesn't necessarily imply joint independence.

Joint Independence

Joint independence is a core concept that goes a step further than pairwise independence. For events \(A, B,\) and \(C\) to be considered jointly independent, they must meet additional criteria. Not only must each pair be independent, but their joint occurrence must also equal the product of their individual probabilities:

• \(P(A \cap B \cap C) = P(A)P(B)P(C)\)

This condition can be tricky. It ensures that knowing two events doesn't affect the occurrence of the third one at all. You can think of it as each event carrying no more influence than it does alone, even when all are considered together. From the example given, analyzing coin toss outcomes, where you have complexities like outcomes dependent on the number of heads or tails, showcases scenarios where joint independence can be achieved even if pairwise independence isn’t present. Events, therefore, can behave independently in unison, yet exhibit dependencies when looked at pairwise, indicating diverse interactions within a probability space.

Probability Space

A probability space forms the mathematical foundation that allows us to discuss abstract terms like independence in a solid framework. A probability space consists of three main components:

• Sample Space \(\Omega\): All potential outcomes.
• Events: Subsets of the Sample Space.
• Probability Measure \(P\): Assigns probabilities to events satisfying certain axioms.

This structure allows us to systematically determine and discuss the likelihood of events, be they dependent or independent. In our example, the sample space \(\Omega = \{1, 2, 3, 4\}\) with \(A, B,\) and \(C\) as events, illustrates probabilities assigned equally to outcomes. The equal probability (\(0.25\) for each event) simplifies computations and helps illustrate pairwise and joint independence further. By setting clear rules within a probability space, it’s easier to explore intricate probabilistic properties and relationships between events, revealing the nuanced nature of independence across different event spectrums.