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When we talk about the probability of winning a series, especially in scenarios with independent trials like athletic games, we often make use of a particular kind of probability distribution known as the binomial probability distribution. This type of distribution helps us to determine the probability of obtaining a fixed number of successful outcomes in a specific number of trials.

Under this distribution, the number of games won by the stronger team, given fixed probabilities for winning individual games, fits perfectly. Here's why: first, we have a sequence of independent events (each game being won or lost), and each event has only two possible outcomes (win or loss). Just as importantly, the probability of winning remains consistent across games.

Moreover, this distribution makes calculating the likelihood of winning an exact number of games straightforward. For instance, if the team needs to win 4 out of 7 games, we use the binomial formula which combines the probability of winning individual games and the number of ways in which these wins can be arranged.

Combinatorics plays a critical role when it comes to calculating probabilities in a series of events. It's essentially the mathematics of counting without having to count each arrangement individually. This becomes essential in our exercise problem where we calculate the probability of winning in exactly 5, 6, or 7 games.

To compute these probabilities, we use a concept called the binomial coefficient, commonly written as \( \binom{n}{k} \), representing the number of ways to choose \( k \) successes out of \( n \) trials. This concept is pivotal because it allows us to account for all possible win combinations that lead to winning the series in a certain number of games. For example, if the team wins in exactly 6 games, there are \( \binom{5}{3} \) ways to distribute these 3 wins among the first 5 games, followed by a win in the 6th game.

We consider the wins and losses of the games to be independent events in probability. Essentially, the outcome of one game does not influence the outcome of another. This characteristic is foundational when determining the probabilities using the binomial distribution.

In our exercise, it's stated that each game is won with a probability of 0.6, independently of the other games. This independence is crucial for the calculations, as it allows us to raise the probability to the power of the number of wins needed and multiply this by the probability of losses raised to the power of the number of losses.

Understanding the concept of independent events is not only important for correctly applying the binomial formula but also for grasping why certain probabilities change (or not) in different scenarios. For instance, whether the series is best-of-7 or a 2-out-of-3, the fact that events are independent underlies the entire probability calculation process.