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Sometimes it is easier to calculate the probability of an event not happening than it is to calculate the probability of it happening. This is where the complementary rule of probability comes into play.

Simply put, the complementary rule tells us that the probability of an event not occurring is equal to one minus the probability of the event occurring. In mathematical terms, if the probability of an event is represented as \(P(A)\), then the probability of the event not occurring is represented as \(1 - P(A)\).

In the context of our exercise, we want to find the probability that a car was manufactured by neither General Motors nor Toyota. We already know the probabilities of them being manufactured by either of these companies. By using the complementary rule, we can subtract these probabilities from one. This will give us the answer we want!

This principle can be very handy, especially when dealing with complex problems, as it often simplifies the calculations or gives you a quick shortcut to the solution.

Probability is a fundamental concept in statistics and mathematics that measures how likely an event is to occur. It’s represented as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. When calculating probability, it’s essential to add up all probabilities related to different possibilities correctly.

In our auto sales example from April 2008, we calculate the probability of a car being manufactured by either GM or Toyota by adding their individual probabilities. Given:

• GM probability is \(0.21\)
• Toyota probability is \(0.18\)

The total probability for a car being manufactured by either one is the sum of these two probabilities: \( P(\text{Either GM or Toyota}) = 0.21 + 0.18 = 0.39 \).

This simple addition tells us that 39% of the cars were manufactured by one of these companies. Once we have this figure, we can use it to determine the probability through complementary probability to figure out how many cars were not built by either company.

Event probability refers to the likelihood or chance of a particular event occurring within a set of possible outcomes. It's helpful to think of probability as a forecast. Just like checking the weather, when you calculate probabilities, you predict the likelihood of events happening.

In the case of this exercise, the event in focus is a car being manufactured by neither General Motors nor Toyota. Calculating individual probabilities of various events helps in understanding the bigger picture.

• If you sum the probabilities of all possible distinct outcomes of an experiment (where occurrences are mutually exclusive and collectively exhaustive), it should always total 1.
• Each outcome's probability must lie between 0 and 1.

Using the event probability for our problem, we focus on the outcome that a randomly chosen automobile is not crafted by the major companies listed. By calculating \(1 - 0.39 = 0.61\), we find out there is a 61% chance that a new car is from another manufacturer altogether. This demonstrates how understanding individual event probabilities helps us derive conclusions about other outcomes in a set.