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Pairwise independence occurs when two events are independent of each other, meaning the occurrence of one event does not affect the probability of the other event happening. However, this does not necessarily imply that all events in a set are independent when considered together.
Consider events A (Heads on the first toss), B (Heads on the second toss), and C (Both tosses result in the same outcome) from the coin toss example:

• Event A: The probability of heads on the first toss, denoted as \( P(A) = \frac{1}{2} \)
• Event B: The probability of heads on the second toss, \( P(B) = \frac{1}{2} \)
• Event C: The probability that both tosses result in the same outcome, \( P(C) = \frac{1}{2} \)

To confirm pairwise independence:
Check that the probability of the intersection of any two events equals the product of their probabilities:

• \( P(A \cap B) = \frac{1}{4} \) matches \( P(A) \times P(B) = \frac{1}{2} \times \frac{1}{2} \)
• \( P(A \cap C) = \frac{1}{4} \) matches \( P(A) \times P(C) = \frac{1}{2} \times \frac{1}{2} \)
• \( P(B \cap C) = \frac{1}{4} \) matches \( P(B) \times P(C) = \frac{1}{2} \times \frac{1}{2} \)

All pairs are independent, confirming that A, B, and C are pairwise independent.
This reveals an interesting aspect of probability: even when events are pairwise independent, they might not be independent altogether.

Joint probability refers to the probability of two or more events occurring at the same time. In the context of the given problem, we calculate joint probabilities to understand how different events overlap when tossing a coin twice.
In our example:

• The joint probability \( P(A \cap B) \) is the probability of obtaining heads on both the first and second tosses (outcome HH), which is \( \frac{1}{4} \).
• The joint probability \( P(A \cap C) \) is the probability of heads on the first toss, and both tosses being the same. This also corresponds to the outcome HH, yielding \( \frac{1}{4} \).
• The joint probability \( P(B \cap C) \) is similar, as it involves heads on the second toss and both being the same, yielding \( \frac{1}{4} \).

These probabilities help us see the interactions between events. When combined, they give us insights into how events relate to each other.
Joint probabilities are important for understanding the connections between events and are foundational for more complex probability calculations.

Conditional probability is the probability of an event occurring given that another event has already occurred. It helps to account for known information, refining our probabilities when conditions change.
For event C being independent of events A and B individually, we evaluated:

• \( P(C|A) = P(C) \), which indicates that knowing A occurs does not change the probability of C.
• \( P(C|B) = P(C) \), suggesting similar independence with B.

Conditional independence tests whether information about one event influences another. However, when we evaluate conditional probability for the conjunction of A and B:

• \( P(C|A \cap B) = 1 \) because having heads on both tosses (HH) ensures that both tosses are the same.
• Comparing this to \( P(C) = \frac{1}{2} \), we see that \( P(C|A \cap B) eq P(C) \), establishing dependence when A and B are considered together.

This examination reveals how dependencies may appear when conditions change or more information is given, illustrating the nuanced nature of probability.