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When we talk about independent events in probability, we refer to situations where the occurrence of one event does not influence the likelihood of another. In our case, the events are a child contracting measles and a child contracting chickenpox in a single year. When events are independent, calculating the combined probability is straightforward: you simply multiply the individual probabilities of each event.
For instance, if a child has a 0.12 chance of catching measles and a 0.2 chance of catching chickenpox, and these are independent events, the probability of a child getting both diseases is the product of the two probabilities, which is 0.12 times 0.2. In mathematical terms, if Event A and Event B are independent, then the probability of both A and B occurring is given by
\[ P(A \text{ and } B) = P(A) \times P(B). \]

When it comes to probability calculations, the process generally involves understanding the type of probability model that applies to the situation at hand—independent, dependent, conditional, or combinatorial. In the measles and chickenpox example, we start by assuming the events are independent as per the exercise prompt and perform a simple multiplication as explained earlier.
Once we have the probabilities for each individual event, we can analyze further, just as we did in the second and third steps of the solution, where we considered given statistical probabilities. Calculating the probability of both diseases requires us to take into account this given data. The calculations reaffirmed our earlier product rule for independent events when we found that the probability of getting chickenpox and then measles is the same as getting measles and then chickenpox, pointing to the independence of the two events.

The intersection of probability and statistics becomes evident when analyzing real-world data to determine the independence or dependence of events. Statistics often provide insights into probability models by showing how often events occur both individually and conjointly. In our exercise, we are presented with statistics showing that the probability of a child getting both measles and then chickenpox is 0.006. This empirical value contrasts with the value calculated under the independent events assumption, guiding us to revisit our initial hypothesis about whether these events are truly independent or whether there's an underlying factor influencing the probability.

Combinatorial probability deals with the likelihood of the occurrence of certain combinations of events, particularly when the events are dependent on each other. While our primary example doesn't delve deeply into combinatory aspects, understanding the concept is crucial when dealing with more complex problems where the order of outcomes or specific groupings becomes significant.
In combinatorial probability, we often use tools such as permutations and combinations to calculate the probability of various scenarios. This is particularly useful in games of chance, in which players might be interested in the likelihood of drawing a specific hand of cards from a deck or rolling a certain combination of numbers on a set of dice. However, in the measles and chickenpox problem, we're looking at a simpler situation, assuming the diseases to be independent and not exploring the combinatory aspects in our calculations.