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In probability theory, independent events play a crucial role in calculating outcomes that involve multiple occurrences, such as our exercise with the network servers. Two or more events are considered independent when the occurrence of one does not affect the occurrence of the other. For example, if you flip a coin and roll a die, the outcome of the coin does not impact the result of the die.

In the context of our server problem, each server's status being down or operational is independent of the others. This means that the likelihood of any one server being operational does not alter the probability that another server is operational. Independence is a key factor because it simplifies the calculation of the overall probability. This simplifies our calculations because it leads to multiplying the probabilities of each independent event.

When addressing multiple independent events, remember:

• Multiply their individual probabilities to find the probability of all events occurring.
• The outcome of one does not influence the others.

Complementary probability is a useful concept when faced with questions about 'at least one' scenarios, like determining if at least one server is operational. The complement of an event is everything that can happen outside of that specific event. So, if we wish to find out the likelihood of an event happening, we also consider its complement - the probability it does not occur.

Let's break it down further. In our server problem, we are interested in the probability of at least one working server. The complementary event would be if none of the servers are operational, i.e., all are down. By calculating this, we can subtract it from 1 to find the likelihood of having at least one operational server.

Here's a simple process to apply complementary probability:

• Identify the event of interest and its complement.
• Calculate the probability of the complement (all unfavorable outcomes).
• Subtract the complement's probability from 1 to get the desired probability.

This approach often makes complex probability problems more manageable and intuitive.

When it comes to calculating probability, it involves determining the likelihood that a particular event will happen out of all possible outcomes. In our scenario with the network servers, the calculation involved determining the probability of none of them being functional and then using the complement to find the probability that at least one is functioning.

The main steps in probability calculations for multiple independent events are as follows:

• Determine the total probability of a single event. For a server being down, it was 0.05.
• For the compound event where all servers are down, multiply the individual probabilities: \[ (0.05)^3 = 0.000125 \]
• Use complementary probability to find the probability of the event of interest:\[ P(\text{at least one operational}) = 1 - P(\text{all down}) = 1 - 0.000125 = 0.999875 \]

The result is the likelihood of having at least one working server. Working step-by-step simplifies complex probability questions, making it much easier to solve problems involving multiple events.