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The concept of a Hypothetical 1000 Table is a useful tool in probability and statistics, helping us visualize data in terms of manageable numbers. Imagine you have 1000 hypothetical customers, and this helps simplify percentage calculations. For example, if 80% of customers subscribe to cable TV, that translates to 800 customers in this hypothetical scenario. Similarly, 42% for Internet translates to 420 customers. This method makes probabilities easier to understand and calculate.
It also becomes simpler to deal with overlapping probabilities. We were told that 97% subscribe to at least one service. The overlap, or both subscribing to cable and Internet, can be calculated by adding the individual numbers and subtracting those accounted twice (in both categories). Doing so gives us 250 customers or 25% of the sample subscribing to both services. This table break down can swiftly be used to calculate other required probabilities. It turns abstract percentages into concrete numbers aiding in better comprehension.

Joint Probability refers to the probability of two events happening at the same time. In the cable company scenario, this means a customer subscribes to both cable TV and the Internet. We calculate joint probability by dividing the number of customers subscribing to both services by the total number of customers.
Joint probability is a key concept, especially when dealing with dependent events. Here, the joint probability is calculated as:

• Number of customers subscribing to both services = 250
• Total customers = 1000

This leads to a joint probability of \(\frac{250}{1000} = 0.25\) or 25%. Understanding joint probability is critical in statistics to analyze how two events correlate with each other and their combined likelihood in any given population.

Events are considered mutually exclusive if they cannot happen at the same time. For example, in our case, subscribing to only cable TV and only Internet are mutually exclusive events, because a customer can only subscribe to one in this context, not both concurrently.
Knowing which events are mutually exclusive helps in determining probabilities, especially when determining 'either-or' scenarios. For instance, the probability of a customer subscribing to exactly one service involved simply adding those subscribing exclusively to either TV or Internet:

• Cable TV only = 550 customers
• Internet only = 170 customers

Thus, the probability of subscribing to exactly one service is \[\frac{550 + 170}{1000} = 0.72\] or 72%. This is because both events don't overlap; one can observe how the probabilities are additive for mutually exclusive events.