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Bayes' theorem is a fundamental formula used to determine the probability of an event, based on prior knowledge of conditions that might be related to the event. It's named after the Reverend Thomas Bayes, who first provided an equation that allows new evidence to update beliefs. To understand the theorem, we start by considering two related events, let's say event A and event B.

For cost overrun scenarios, this theorem is particularly useful. It helps us to revise the probability of a specific consulting firm being responsible for a cost overrun given that we have observed that a cost overrun has indeed occurred. In mathematical terms, if we want to find out the probability of event A (choosing firm C, for example) given that event B (a cost overrun) has occurred, Bayes' theorem gives us the formula:
\[ P(A|B) = \frac{{P(B|A) \times P(A)}}{{P(B)}} \]
Here, \(P(A|B)\) is the probability of event A given B has occurred, \(P(B|A)\) is the probability that event B occurs given A is true, \(P(A)\) is the probability of event A, and \(P(B)\) is the probability that event B occurs, also known as the total probability. This theorem provides a way to update our probabilities based on new information, which is an incredibly valuable tool not only in statistics but also in various real-world decision-making processes like economic forecasting, medical diagnosis, and even machine learning algorithms.

Conditional probability is simply the probability of one event occurring with some relationship to one or more other events. For example, it answers questions such as, 'What is the probability that a consulting firm will experience a cost overrun, given that it has been hired by a specific federal agency?'

Mathematically, the conditional probability of event A occurring given that event B has occurred (denoted as \(P(A|B)\)) can be calculated using the formula:
\[ P(A|B) = \frac{{P(A \text{ and } B)}}{{P(B)}} \]
where \(P(A \text{ and } B)\) refers to the probability of both events A and B occurring together.

It's crucial to differentiate between conditional probability and the probability of two independent events both happening (denoted by multiplication of their separate probabilities). Conditional probability takes into account how the outcome of one event affects the outcome of another. This concept is fundamental to Bayes' theorem, as it provides the means to calculate the likelihood of an event based on the occurrence of another related event.

The total probability is a rule that allows one to calculate the probability of a certain event, based on the probabilities of different scenarios leading up to it. It's the bedrock upon which we base our calculation for Bayes' theorem, as it encapsulates all the possible ways an event can occur and sums their weighted probabilities.

In our consulting firm example, we can calculate the total probability of a cost overrun by considering all scenarios—each firm being hired—and their individual probabilities of causing a cost overrun. The formula for the total probability of event B (a cost overrun), assuming that the occurrence of B is dependent on a prior event A (choosing a specific consulting firm), is:
\[ P(B) = P(A_1)P(B|A_1) + P(A_2)P(B|A_2) + \text{...} + P(A_n)P(B|A_n) \]
where \(A_1, A_2, ..., A_n\) are all the different scenarios or events that can occur before B.

The principle of total probability is very powerful, especially when we have a series of events or conditions that lead to an outcome. It helps us account for all possible ways something can happen, which is crucial when predicting the likelihood of outcomes in complex systems such as weather, finance, and project management.