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In probability theory, understanding whether events are independent is crucial for calculating combined probabilities accurately. Two events are considered independent if the outcome of one event does not influence the outcome of the other.
This means that the probability of both events happening is simply the product of their individual probabilities.
Imagine flipping a coin twice. The outcome of the first flip (heads or tails) doesn't affect the outcome of the second flip. Similarly, in this exercise, whether the first person owns a dog does not influence whether the second person owns a dog. Therefore, the events are independent.
This allows us to multiply their probabilities directly when calculating the chance that neither of two randomly picked individuals own a dog. This fundamental principle makes many real-world probability calculations possible.

Probability calculation is a way to quantify the likelihood of an event occurring. Probabilities range from 0 to 1, where 0 means an event is impossible, and 1 means it is certain. The probability of an event can be expressed using the formula:\[ P(A) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}} \]For example, if 21% of people own dogs, we can say the probability of a randomly selected person owning a dog is 0.21.
Consequently, the probability of the complementary event, i.e., not owning a dog, can be found using another basic probability formula:\[ P(\text{not owning a dog}) = 1 - P(\text{owning a dog}) = 1 - 0.21 = 0.79 \]This approach helps us understand and solve many basic probability problems efficiently.

Complementary probability is a useful concept when dealing with probabilities that add up to 1. For any event, its complement is the event that the original event does not occur. The sum of their probabilities is always equal to 1.
Consider the situation where 21% of people own dogs. The complement of this event is that a person does not own a dog, which we calculated as 79% since:\[ P(\text{not owning a dog}) = 1 - P(\text{owning a dog}) = 1 - 0.21 = 0.79 \]Complementary probability helps simplify complex probability problems, especially when involved in multiple calculations.
By understanding the complementary probabilities, we can work backwards from what is not happening to deduce what is happening, offering a straightforward way to ensure our calculations cover all possibilities.