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Independent events are a fundamental concept in probability. When two or more events are independent, it means that the outcome of one event does not affect the outcome of the other(s). In other words, the events occur without influencing each other.
To determine if events are independent, consider this: the probability of the events happening together can be found by multiplying their individual probabilities.

• For example, if the probability of event A occurring is \(P(A)\) and the probability of event B occurring is \(P(B)\), the probability of both events occurring is \(P(A) \times P(B)\).
• In the case of Copykwik's photocopy machines, each machine breaking down on a single day is considered independent of the others.

This means that the failure of one machine does not cause or influence another machine to fail. Thus, these events are perfectly qualified as independent events.

In the context of this exercise, a machine breakdown refers to a photocopy machine no longer operating on a given day. For each photocopy machine at Copykwik, there is a given probability that it will cease to function on any particular day.
Each machine has its own specified probability of breakdown:

• Machine A: \(\frac{1}{50}\)
• Machine B: \(\frac{1}{60}\)
• Machine C: \(\frac{1}{75}\)
• Machine D: \(\frac{1}{40}\)

These probabilities are independent from one another, meaning the likelihood of one machine breaking down does not change because another machine may or may not break down. Understanding these individual probabilities is crucial when trying to determine the probability of an event such as all machines breaking down simultaneously or not breaking down at all.

Event probability refers to the chance or likelihood of a specific event occurring. Each event in probability is assigned a value between 0 and 1.
A probability of 0 indicates an event is impossible, while a probability of 1 means it is certain. You can think of probabilities as fractions, decimals, or percentages to understand how likely an event is to occur.

• An individual machine breaking down can be considered an event, and its probability has been predefined for each machine based on its performance data.
• When assessing multiple events, like all machines breaking down, the overall probability is determined by the relation between these individual events.
• In this exercise, we're interested in the event probability for a combination or absence of events: all four machines breaking down together, and none of them breaking down at all.

This concept helps us predict occurrences or plan for maintenance at Copykwik.

Probability calculation involves the mathematical steps used to determine the probability of events. Calculating probability becomes particularly interesting when dealing with multiple events.
To calculate the probability of all independent events occurring together, multiply the individual probabilities:

• Using the given exercise example, for all machines breaking down, we compute \(P(A \cap B \cap C \cap D) = \frac{1}{50} \times \frac{1}{60} \times \frac{1}{75} \times \frac{1}{40} = \frac{1}{9{,}000{,}000}\).

Similarly, to calculate the probability that none of the machines break down, first determine the probability of each machine not breaking down:

• For example, \(P(A') = 1 - \frac{1}{50} = \frac{49}{50}\).

Then multiply these results to find the combined probability:\[P(A' \cap B' \cap C' \cap D') = \frac{49}{50} \times \frac{59}{60} \times \frac{74}{75} \times \frac{39}{40} = \frac{167{,}965}{180{,}000}\].
This approach ensures precision and clarity when predicting the likelihood of compound events.