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The probability mass function (PMF) is a fundamental concept in probability theory, particularly when dealing with discrete probability distributions like the Poisson distribution. At its core, the PMF provides you with the probability that a discrete random variable is exactly equal to a specific value.
In our exercise, the Poisson PMF is used to figure out the probability that any specific number of calls, say \( k \), occurs in one minute. This is represented mathematically as:

• \( P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!} \)

Here, \( \lambda \) is the average rate (5 calls per minute in this case), and \( k \) is the number of calls we're looking to find the probability for, such as 0, 1, 2, and so on.
The PMF helps break down the scenario into understandable parts, specifying how likely it is for different numbers of calls to occur within one minute. This builds a detailed landscape of probabilities for each possible count of calls, crucial for further calculations of cumulative and complementary probabilities.

Cumulative probability involves the addition of probabilities from the probability mass function to find the likelihood of a random variable being less than or equal to a certain value. It's an essential concept for determining probabilities of "at most" scenarios.
For example, in our exercise, we calculate the cumulative probability of having 10 or fewer calls in a minute. Mathematically, it's presented as:

• \( P(X \leq 10) = \sum_{k=0}^{10} P(X = k) \)

This means adding together probabilities for 0 calls, 1 call, up to 10 calls. This step is key to knowing how often the switchboard can handle the calls without being overtaxed.
Calculating cumulative probability gives us insights beyond individual outcomes and prepares the ground for finding the complement probability. By summing up the probability of all scenarios up to a certain point, it provides a broader view of how the variable behaves across multiple events.

Complement probability refers to finding the likelihood of the opposite outcome of an event. If you know the probability of an event happening, the complement tells you the probability of the event not happening. It's an easy way to find out what's left over.
In our exercise, we're interested in finding the probability that more than 10 calls happen in a minute. Instead of calculating directly, we use the cumulative probability and subtract it from 1:

• \( P(X > 10) = 1 - P(X \leq 10) \)

This formula is straightforward because knowing the cumulative probability of 10 or fewer calls allows us to quickly figure out the rest by finding what we have left of the whole probability space (which totals to 1).
Complement probability efficiently leverages the work done to calculate the cumulative probability, giving a simple solution to our original problem of estimating the probability of the switchboard being overtaxed.