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Complementary probability is an important concept in probability theory. It involves calculating the likelihood of an event not happening by subtracting the probability of the event from one. This is useful when it's easier to calculate the probability of an event not happening than the event itself.
For instance, in our original exercise, we used complementary probability to find the likelihood of at least one floodlight remaining functional. We know the probability of a single floodlight not burning out is 0.99, hence the probability of all floodlights burning out can be expressed as \((0.99)^n\).

• Complementary probability helps simplify problems by focusing on the opposite event.
• It is denoted by \(1 - P(A)\) where \(P(A)\) is the probability of the event occurring.

Understanding complementary probability makes it easier to solve problems where the desired outcome is the inverse of the direct calculation.

Logarithms are mathematical functions that help solve equations involving exponential terms. They translate multiplication into addition, making calculations more manageable. In the context of our problem, logarithms were used to isolate the number of floodlights, \(n\), when solving for the inequality:1. Convert the exponential expression to a logarithmic one.2. Use logarithm properties to simplify computations.For example, we used:

• \(\log((0.99)^n) = n \cdot \log(0.99)\)
• To solve the inequality, \((0.99)^n \le 0.00001\), it becomes \(n \ge \frac{\log(0.00001)}{\log(0.99)}\)

Using a calculator, this converts complex multiplication into simple division, showing how useful logarithms can be in practical scenarios.

An inequality is a mathematical statement that compares quantities, showing how one quantity is less than or greater than another. In this exercise, the inequality \(1-(0.99)^n \ge 0.99999\) was crucial to find the minimum number of floodlights.The steps involved:

• Transform the expression to make \(\le\) by rearranging terms.
• Isolate the variable of interest, in this case, \(n\), using logarithms.
• Find the smallest integer that satisfies the inequality, which is \(n = 1163\).

Inequalities allow us to determine bounds and ranges, rather than exact values, offering practical solutions for real-world problems.

Events are termed independent if the occurrence of one doesn't affect the other. In this exercise, each floodlight functions independently, meaning the probability that one floodlight burns out doesn’t impact any other.Key characteristics of independent events:

• The probability of both events occurring is the product of their individual probabilities.
• Independence simplifies calculations because the impact of one event doesn’t alter another event.

For the floodlight scenario, since each operates independently, the probability calculation becomes simplified, using \((0.99)\) for each light over the span of the year: \((0.99)^n\) shows the compound effect of \(n\) lights possibly burning out. Thus, understanding independence helps effectively tackle this and similar problems.