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A binomial distribution is a probability distribution that summarizes the likelihood that a value will take one of two independent states and is used when there are two possible outcomes. Each outcome is referred to as a "trial." This distribution is applicable when trials are repeated a fixed number of times, each trial is independent, and the probability of the outcome remains constant throughout the trials.
For instance, consider the problem of a hotel accepting reservations. Here, a reservation either shows up or doesn’t show up, which forms a classical scenario of a binomial distribution.

• Number of trials (n): This represents the number of reservations the hotel accepted, which is 215.
• Probability of success (p): The probability that a guest with a reservation shows up. In this case, it is 90% or 0.9, as 10% do not show up.

This is a perfect setup for applying binomial distribution to understand the likelihood of having a certain number of guests claiming their reservations.

The complementary probability principle is a powerful tool in calculating probabilities for events, particularly when the direct calculation of the desired event is complex or inconvenient. It revolves around the idea that the probability of an event plus the probability of its complement equals one.
In our scenario, to simplify the mathematics, we utilize the complementary probability. Instead of directly calculating the probability of 200 or fewer guests showing up (P(X ≤ 200)), we calculate the probability of more than 200 guests showing up (P(X > 200)) and then subtract it from 1.
Mathematically, this means:

• \( P(X ≤ 200) = 1 - P(X > 200) \)

This essentially helps to consider only the specific cases where we have issues, i.e., more guests than rooms, making it already significant in terms of assurance.

The probability mass function (PMF) is a fundamental concept when dealing with discrete random variables such as those described by a binomial distribution. It provides the probabilities of each possible outcome occurring.
In our hotel problem, the PMF allows us to calculate the probability of exactly \( k \) guests claiming their reservation out of the \( n \) reservations, given a chance \( p \) of each guest showing up. The PMF for a binomial distribution is formulated as:
\[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \]

• \( \binom{n}{k} \) is the binomial coefficient, which determines the number of ways \( k \) successes can occur in \( n \) trials.
• \( p^k \): Probability of \( k \) guests showing up.
• \( (1-p)^{n-k} \): Probability of the remaining guests not showing up.

In the given problem, this function will help us calculate the probability for each possible number of arrivals over 200, from 201 to 215. The solution is obtained by summing these individual probabilities to find the total probability of more than 200 arrivals and thus the complementary probability for 200 or fewer arrivals.