91Ó°ÊÓ

In probability, independent events are those events where the outcome of one event does not affect the outcome of another. This is an important concept to grasp when calculating probabilities for multiple scenarios.
Imagine you have a bag of marbles. If you pull out a marble, note its color, then return it before drawing another, each draw is independent. The result of the first draw doesn't change the chances for the second one. This is similar to our exercise, where selecting the first person who thinks they are lucky doesn't change the odds for selecting the next person.
For our given problem, selecting 3 people with the probability that each considers themselves lucky is an independent event. This means each person's belief in luck doesn't linger or influence the next. The independence is crucial when calculating the combined probabilities, like finding out the chance that none are considered lucky.

• Probability for one person being lucky = 0.12
• Probability for one person not being lucky = 0.88

The complement rule is a useful tool in probability that helps calculate the likelihood of an event not occurring. In simple terms, if you know the probability of an event happening, the complement rule helps you find the probability it doesn't happen.
The rule works as follows: If an event has a probability of occurring, its complement is 1 minus that probability. This makes sense because probability must add up to 1 (or 100%). Thus, if the probability of event A occurring is \( P(A) \), then the probability of it not occurring is \( P(\text{not }A) = 1 - P(A) \).
In our exercise, we wanted to know the probability of at least one person believing they are lucky. Understanding the complement rule allowed us to first calculate the probability of none of the 3 people considering themselves lucky and then use this result to find the probability that at least one does. By subtracting from one, we focus on the broader scenario and simplify the complex calculations.

Raising a probability to a power is a common technique used in probability calculations when dealing with multiple independent events. This is often seen as: \( (\text{probability of an event})^n \), where \( n \) is the number of times the event occurs independently.
In our scenario, we have the probability of an individual thinking themselves lucky as \( 0.12 \), and likewise, not considering themselves lucky as \( 0.88 \). If you want to calculate the probability of more than one person being unlucky in a group, you raise the probability of a single event happening to the power of the number of people in the group.
This can look daunting, but it can be simplified by:

• Probability of none being lucky = \( 0.88^3 \)
• Simplifies complex event calculations

By using this method, you are breaking down a complex situation into smaller, more manageable parts. Calculating \( 0.88^3 \) gave us the probability of none being lucky, making it possible to then use the complement rule for finding out when at least one person considers themselves lucky.