91Ó°ÊÓ

In probability, the Complement Rule is a tool that can make solving complex problems simpler. For any given event, its complement includes all outcomes that are not considered in the original event.
This rule can be applied by calculating:

• The probability of the complement being: \[ P( ext{complement}) = 1 - P( ext{event}) \]

For example, if the probability that at most one customer buys an electric dryer is 0.428, then the probability of at least two customers purchasing an electric dryer (which is the complement of the first event) is:

• \[ P( ext{at least two purchase electric}) = 1 - 0.428 = 0.572 \]

This demonstrates how the Complement Rule helps in determining the probability of an alternate scenario quickly and easily.

Mutually exclusive events are situations in probability where the occurrence of one event makes it impossible for another event to happen at the same time. In simpler terms, these events cannot occur simultaneously.
A classic way to remember this concept is that mutually exclusive events have no overlap.
As an example, consider the scenario where all five customers purchase either gas dryers or electric dryers. The events \( P( ext{all five gas}) \) and \( P( ext{all five electric}) \) are mutually exclusive, meaning it is impossible for all buyers to choose gas dryers and electric dryers at the same time.
Here, both events add up to circumstances where neither holds a mix, thus:

• \[ P( ext{all gas}) + P( ext{all electric}) \]

The sum of probabilities of all mutually exclusive events is less than or equal to 1, because these events cannot cover an instance where both occur simultaneously.

Event probability measures the likelihood of a specific event happening within a given experiment or process. Each event has a probability value between 0 and 1; a probability of 0 means the event will not occur, and a probability of 1 means the event is certain to occur.
When evaluating complex situations involving multiple types of events, such as the combined possibility of different dryer purchases, understanding event probability is crucial. In our exercise, different scenarios, like all dryers being gas, all being electric, or having a mix, each have calculated probabilities:

• All gas dryers: \( P( ext{all five gas}) = 0.116 \)
• All electric dryers: \( P( ext{all five electric}) = 0.005 \)
• At least one of each type: \[ P( ext{at least one of each}) = 1 - P( ext{all gas}) - P( ext{all electric}) = 0.879 \]

Understanding event probability helps one make sense of the likelihood of various complex outcomes, enabling informed predictions and decision making. By calculating the probabilities of different event scenarios, we derive insights into what is most likely to happen given specific conditions.