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In probability theory, independence of events refers to a situation where the occurrence of one event does not impact the occurrence of another. For example, in the context of a retail store, consider a customer’s choice of payment between cash and credit card, and whether they purchase an extended warranty. These choices are independent if buying an MP3 player with cash does not influence the decision of purchasing an extended warranty and vice versa.

Mathematically, this is expressed as:

• Two events, say event A and event B, are independent if \( P(A \cap B) = P(A) \times P(B) \).

In our example, \( M \) represents paying by cash, and \( E \) represents purchasing an extended warranty. Given \( P(M) = 0.47 \) and \( P(E) = 0.16 \), they are independent because \( P(M \cap E) = 0.47 \times 0.16 \). This simplifies the analysis and calculations significantly, as knowledge about one event doesn't change the expectation of another.

Joint probability involves finding the likelihood of two events happening at the same time. For instance, the probability of paying with cash and purchasing an extended warranty would be a joint probability. This is seen in the intersection of two events, mathematically described by \( P(A \cap B) \).

In our original exercise, the joint probability was calculated for cash payment and extended warranty as \( P(M \cap E) = 75.2 \approx 75 \). This means that out of 1000 hypothetical customers, around 75 are expected to pay by cash and also purchase an extended warranty. Understanding joint probability helps in figuring out how two independent events coincide or overlap with each other in a given scenario.

The union of events in probability theory is expressed as the probability that at least one of multiple possible events occurs. This is denoted by \( P(A \cup B) \), meaning event A happens or event B happens, or both occur. It’s a central concept when determining the probability of multiple scenarios happening together.

In the exercise provided, finding \( P(M \cup E) \) refers to calculating the likelihood that a customer either pays with cash, purchases an extended warranty, or does both. This is also known as the "or" probability. To calculate this, use the formula

• \( P(M \cup E) = P(M) + P(E) - P(M \cap E) \)

Substituting the given probabilities, \( P(M \cup E) \) is found to be 0.57. Thus, in our "hypothetical 1000" table scenario, out of 1000 customers, we expect about 570 to either pay in cash, buy an extended warranty, or do both. Understanding the union of events is crucial for assessing aggregate probabilities.