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The General Multiplication Rule helps us understand the probability of two events happening together. It's a foundational concept in probability and statistics. Simply put, it allows us to find out how likely it is that both Event A and Event B occur together by using the formula:

• \( P(A \cap B) = P(A) \times P(B|A) \).

In this formula, \( P(A) \) is the probability of the first event, and \( P(B|A) \) is the probability of the second event happening given that the first event has occurred.
In our gym example, we first know that 8% of adults belong to health clubs. So, \( P(A) = 0.08 \). Then, given that someone belongs to a club, 45% of them go at least twice a week, meaning \( P(B|A) = 0.45 \).
By following the General Multiplication Rule, you can calculate the probability of an adult belonging to a gym and going at least twice a week as \( P(A \cap B) = 0.08 \times 0.45 = 0.036 \). This helps us realize that the joint probability of both events happening is 3.6%.

Conditional probability refers to finding the probability of an event occurring when we know that another event has already happened. It's shown in the form \( P(B|A) \), which reads as "the probability of B given A." This approach makes it easier to calculate complex probability scenarios by narrowing down the sample space based on the already known event.
In the context of our exercise, suppose we first know that an adult is already a member of a health club—this is our event A. Now, we want to find the probability that this specific adult goes to the club at least twice a week, which is event B given A.
This probability is written as \( P(B|A) = 0.45 \), indicating that 45% of health club members go to the club at least twice a week. Conditional probability helps us understand interdependent events, enriching our insights into probability analysis in real-world situations.

Problem-solving in statistics often involves breaking down information and applying it within the framework of probabilistic rules and formulas. Understanding how to define events and apply rules such as the General Multiplication Rule is crucial in interpreting data accurately.
Let's revisit our example: the problem defines two events:

• Event A: An adult belongs to a health club (\( P(A) = 0.08 \)).
• Event B: An adult goes to the club at least twice a week given they belong to the club (\( P(B|A) = 0.45 \)).

To solve for the percentage of adults who go to the gym at least twice a week, you multiply the probabilities by using the general multiplication rule \( P(A \cap B) = P(A) \times P(B|A) \).
This structured approach allows step-by-step analysis and helps simplify complex scenarios, cultivating skills that are essential in diverse areas like science, business, and daily decision-making.