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Understanding probability calculations is essential when dealing with random events, like renting apartments. In this scenario, we are interested in the probability of apartments being unrented. Our focus is on two types of probability: *exact* and *cumulative*. 

The probability of a specific outcome, such as exactly 4 apartments remaining unrented, involves straightforward calculation using the binomial distribution. The formula is given by:

• Binomial probability:

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}\]Here, *n* is the number of trials (20 apartments), *k* is the number of successes (unrented apartments), and *p* is the probability of a single success (0.10 for an apartment not rented).

This calculation helps us find how likely it is for exactly 4 apartments to remain unrented. It's done by substituting the appropriate values into the formula. It's a single, distinct probability value that applies only when there are precisely 4 unrented units.

In scenarios involving multiple possible outcomes, cumulative probability provides a comprehensive perspective. It lets us understand the likelihood of a range of events occurring. This is especially valuable when determining the chance of different possible numbers of unrented apartments.

When calculating cumulative probability, we are considering all possible events up to a certain point. In this case, it's about finding the probability of having 4 or fewer unrented apartments:

• Collective Probability Equation:

\[P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)\]

Each term represents a possible outcome with a distinct probability, and adding these probabilities gives us the cumulative probability!

• Significance: For the landlord, this gives a better picture of any scenario that could ensure expenses are covered since 16 or more rented apartments will suffice.

This cumulative perspective aligns better with the goal of meeting expenses as it factors in every possible way to rent out 16 or more apartments.

In this context, independence in probability means that each apartment's rental status is not affected by others. Imagine flipping a coin: each flip doesn’t influence the next. It’s the same here!

The independence assumption is pivotal for applying the binomial distribution within this exercise. Because each apartment's rental status is independent, we use the probability of a single event (an apartment not rented) across multiple, separate trials (each apartment).

This ensures each calculation is isolated from results of other apartments, thus simplifying calculations and maintaining accuracy in predicting outcomes. So, when the landlord assesses rental probabilities, they're relying on this key assumption about independence.

Without such independence, results would skew since each event could then be intertwined, requiring more complex probability models. For the sake of simplicity and accuracy, independence here keeps calculations manageable!