On any day, the probability that a person is absent due to illness is \(0.001\). In a workforce of 600 people, calculate the probability that on any day the number of people absent is (a) none, (b) one, (c) more than one, (d) less than three.

Probability theory is the branch of mathematics that deals with the likelihood of different outcomes in random events. The primary goal is to quantify the chance of various results, providing a numerical representation of uncertainty. In the context of a workforce potentially being absent due to illness, the probability that any single person is absent on a given day is a real-life application of this theory.

Formally, probability can be calculated using the following formula: \( P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \). In our exercise, the 'favorable outcome' is the specified number of people being absent, and the 'possible outcomes' encompass all the ways that scenario can occur among the 600 employees.

Improving the understanding here involves recognizing that the 'favorable outcomes' depend on two conditions: the probability of a person being absent and the probability of them attending. When we calculate probabilities for several people, those conditions are combined multiplicatively, reflecting how independent events interact in probability theory.

The binomial theorem is a cornerstone of algebra that provides a systematic method for expanding binomial expressions raised to any power. In the realm of probability, this theorem underlies the binomial distribution, which is used when there are only two possible outcomes (like the presence or absence of employees due to illness) for each trial, and each trial is independent.

The formula for the binomial probability of exactly \(k\) successes out of \(n\) trials is: \( P(k;n,p) = \binom{n}{k}p^k(1-p)^{n-k} \), where \(p\) is the probability of success on any given trial, \(k\) is the number of successes, and \((1-p)\) is the probability of failure.

Understanding how to use the binomial theorem in probability calculations involves combinatorics to determine \(\binom{n}{k}\), the number of ways to choose \(k\) successes out of \(n\) trials, multiplied by the probability of that specific combination of successes and failures. This is precisely what we apply in the exercise when computing the probabilities of zero, one, more than one, and less than three absentees.

Combinatorics is the branch of mathematics focusing on counting, combining, and arranging sets of elements. In probability, particularly in solving binomial distribution problems, combinatorics is essential for figuring out the number of ways events can occur.

The binomial coefficient, represented as \(\binom{n}{k}\), is a fundamental combinatoric concept used in the exercise. It denotes the number of ways to choose \(k\) items from \(n\) without regard to order. Calculating the binomial coefficient involves factorials, where the factorial of a number \(n\), denoted by \(n!\), is the product of all positive integers up to \(n\).

The explicit formula for the binomial coefficient is \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\), showing the relationship between combinations and probabilities. The understanding of combinatorics allows us to handle more complicated scenarios where the events are numerous and varied, as in the workforce absence example. It's important to grasp that these mathematical combinations provide the means to account for every possible arrangement of absent and present employees that lead to the outcomes we're interested in.