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Cumulative probability is a concept that helps us determine the likelihood of a random variable being less than or equal to a certain value. It provides a way to consider the accumulated probabilities of multiple outcomes, rather than focusing on one specific result. For instance, if you want to know the probability of having 3 or fewer robberies in a day, cumulative probability allows you to sum up the probabilities of 0, 1, 2, and 3 robberies.

Cumulative probabilities are typically given in probability tables, which offer quick references. In practical applications, finding this probability saves time and ensures calculations are accurate because they factor in several individual probabilities in one step. This concept is especially useful in the Poisson distribution when determining probabilities within specific constraints such as 'at most' or a certain range.

The complement rule is a fundamental concept in probability theory that allows you to find the probability of an event by using its opposite. Specifically, it states that the probability of an event not occurring is 1 minus the probability of the event occurring.

In the context of our problem, to find out the probability of having at least 12 robberies, we would first calculate the cumulative probability of having 11 or fewer robberies. Then, subtract this from 1 to get the desired probability for at least 12 occurrences. This rule is especially helpful when direct calculations for an event are complex, but its opposite is easier to find.

Probability tables are powerful tools for simplifying the process of finding cumulative probabilities, especially with complex distributions like Poisson. These tables contain pre-calculated probabilities for different values of \( \lambda \), the average rate of occurrence, and k, the number of successes.

In using these tables for Poisson distributions, you can quickly find probabilities for scenarios like 'at most' or 'at least' occurrences by referencing cumulative values. For example, if you want to know the likelihood of having at most 3 robberies, the cumulative value for 3 or fewer robberies will provide this answer instantly. These tables are included in many textbooks and statistical appendix sections for rapid access and accuracy.

The Poisson formula is a critical mathematical tool used to find the probability of a given number of events happening within a fixed interval of time or space. It is especially useful when events occur independently and at a constant average rate. The formula is given by:\[P(X=k)= \frac{e^{-\lambda} \cdot \lambda^{k}}{k!}\]where \( \lambda \) is the average number of occurrences, \( e \) is Euler's number (approximately 2.718), \( k \) is the number of events we're interested in, and the exclamation mark "!" denotes the factorial.

In our specific robbery example, you would plug in \( \lambda = 6.3 \) and \( k=3 \) to calculate the probability of exactly 3 robberies occurring on a given day. The versatility and straightforwardness of the Poisson formula make it a go-to in scenarios where you need to model the probability of events over time or space.