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In combinatorics and probability, independent events are those whose outcomes do not affect each other. This means the occurrence of one event has no impact on the likelihood of another event happening. In our exercise about automobile orders, each choice regarding transmission, air conditioning, and color is considered an independent event.

• Choosing between automatic or standard transmission does not influence the options available for air conditioning or color.
• Similarly, selecting to have or not have air conditioning does not affect the transmission or color choices.
• The selection of a color for the automobile cannot change the types of transmission or the presence of air conditioning.

Because these choices are independent, they can be made separately, one after the other, without previous choices affecting subsequent options. This independence allows us to use the multiplication principle to determine the total number of combinations possible.

Probability is the measure of the likelihood that an event will occur. It is calculated by dividing the number of favorable outcomes by the total number of possible outcomes. According to the exercise, if all options are available equally, each car configuration has an equal probability of being ordered.

This means that given 16 possible unique configurations, the probability of choosing any specific configuration, like an automatic transmission with air conditioning in red, is:

• Number of favorable outcomes: 1 (since you are picking a specific combination)
• Total number of possible outcomes: 16

Thus, the probability for any single configuration is calculated as:\[\text{Probability} = \frac{1}{16}\]This represents a 1 in 16 chance of any single configuration being chosen if all configurations are equally preferred and there are no other factors influencing the decision.

The multiplication principle is a fundamental concept in combinatorics, which helps to find the total number of outcomes for a series of independent events. When calculating possible configurations for the automobile, each choice represents a stage in the decision process.

• The first choice is between 2 transmission types: automatic or standard.
• This is followed by selecting among 2 types of air conditioning: with or without.
• Finally, the buyer can choose from 4 color options.

To determine the total number of configurations, multiply the number of options available at each stage:\[2 \text{ (transmissions)} \times 2 \text{ (air conditioning)} \times 4 \text{ (colors)} = 16 \text{ possible orders}\]Each decision is multiplied because each is independent from the others, and the total number calculates the comprehensive set of all combinations. This approach allows ordering processes to be streamlined and solutions to complex combinations of events to be simplified.