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Understanding the complement rule in probability is like learning a shortcut for certain types of calculations. Essentially, the complement rule states that the probability of an event not occurring is equal to one minus the probability that it does occur. This becomes particularly handy when it's easier to calculate the chances of an event not happening than the event happening itself.

For example, in the exercise provided, the probability that at least one out of 15 homeowners has earthquake insurance is more easily found by first calculating the probability that none have insurance, and then using the complement rule. To put it into a formula, if we denote the probability of at least one homeowner having insurance as P(at least one), and the probability of none having insurance as P(none), the rule can be expressed as:

\[\begin{equation}P(at least one) = 1 - P(none)\[\begin{equation}
Applying this to our scenario, if P(none) = \((9/10)^{15}\), then P(at least one) is simply 1 minus this value. This approach often simplifies the calculation process in binomial probability problems.

When we talk about the mean of a binomial distribution, we're referring to the expected value or average number of successes in a given number of trials. The mean is calculated as the product of the number of trials (\(n\)) and the probability of success on any given trial (\(p\)).

\[\begin{equation}\text{Mean}, \(mu\) = n * p\[\begin{equation}
In the context of the exercise, where homeowners are randomly chosen for interviews about earthquake insurance, if \(n = 15\) (the number of homeowners interviewed) and \(p = 1/10\) (the probability of a homeowner having insurance), then the mean number of insured homeowners in our sample is \(15 * \frac{1}{10} = 1.5\). This is the 'average' number of homeowners we'd expect to have insurance if we repeated this sampling process many times.

The standard deviation of a binomial distribution measures how spread out the values (in this case, the number of successes) are from the mean. In simpler terms, it tells us how much variability we can expect from the average. For a binomial distribution, the standard deviation (\(\sigma\)) can be calculated using the square root of the product of the number of trials (\(n\)), the probability of success (\(p\)), and the probability of failure (\(1-p\)).

\[\begin{equation}\sigma = \sqrt{n * p * (1-p)}\[\begin{equation}
For our exercise, with \(n = 15\) and \(p = 1/10\), the standard deviation would be \(\sqrt{15 * \frac{1}{10} * \frac{9}{10}}\). This value helps us understand the extent of fluctuation to expect in the number of homeowners with insurance. For instance, if the standard deviation is large, it indicates a high degree of variability around the mean, which might suggest that our sample could be quite different from the general population.