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When dealing with probability, independent events are scenarios in which the outcome of one event doesn't affect the outcome of another. Imagine flipping a coin and rolling a die at the same time. What you get with your coin flip doesn’t alter what number appears on the die.

In the context of the traffic light systems, each system operates independently. This means that whether one system works properly or not has no bearing on how the others perform. Armco's systems, therefore, can be thought of as independent events, each having the same probability of working correctly.

• They are unaffected by the operation status of one another.
• The independence is crucial to applying specific probability rules.

Understanding independent events is essential as it allows for using the multiplication rule in probability calculations.

The multiplication rule is a straightforward yet powerful tool in probability. It applies when you want to find the likelihood of two or more independent events all happening. This rule says: if you have events that are independent, the probability that they all occur is simply the product of their individual probabilities.

Think about it like stacking blocks. Each block (event) stands on its own without leaning on another. In our scenario with Armco's traffic lights, each system has a 95% chance (0.95 probability) of working properly for at least 3 years. To find the probability of all four systems working correctly for three years, you multiply their probabilities together:

• a: For the first system, it's 0.95
• b: For the second system, it's another 0.95
• c: The third system also is 0.95
• d: Finally, the fourth system is still 0.95

So, the combined probability is: \[0.95 \times 0.95 \times 0.95 \times 0.95 = 0.95^4\]

The act of computing the probability of several independent events relying on the multiplication rule is often called probability calculation. It involves multiplying the probabilities of individual events.

• In the original problem, you learned that each Armco system functions properly with a probability of 0.95.
• The goal was to find the probability all four systems perform correctly over three years.

So, to determine this, you multiplied the probability of one event by itself three more times, because it's independent:\[0.95^4 = 0.8145\]

Hence, the likelihood that all four systems will last three years without failing is approximately 81.45%. This showcases the importance of understanding and applying the correct probability rules.

Picture this: you've just become a city planner and your latest project is ensuring that all new traffic lights in your town function correctly for at least three years. This type of probability scenario requires determining how likely it is for multiple components to work over a set period.

For Armco's systems, the initial scenario specifies that 95% of these will function properly for at least three years. The task was to figure out the chance that, when buying four, every single one would work perfectly.
This exercise:

• Involved considering each system as an individual, independent component.
• Necessitated calculating the combined probability using the multiplication rule.

By doing these calculations, you can prepare for real-world situations like managing city infrastructure efficiently. So remember, these scenarios help build your foundational skills in probability and preparedness for planning.