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The concept of sample space is essential when studying probability because it represents all possible outcomes in a given situation. In our exercise, the sample space is comprised of the 3000 families living in the apartment complex. When we talk about sample space, we are considering all these families as potential outcomes when randomly choosing one family.
Understanding the sample space is crucial for calculating probabilities as it lays the foundation for determining the likelihood of specific events. By defining the sample space, you ensure that you have accounted for every possible outcome, making your calculations and resulting conclusions accurate. In this context, our sample space is 3000 families, from which we will calculate the probability of choosing one that paid income tax last year.

When calculating probability, the term 'successful outcomes' refers to the number of times the desired event occurs within the sample space. It's important to first understand what constitutes a success in your scenario. In our exercise, a successful outcome is a family that did pay income tax last year.
This topic is easy to comprehend with a simple subtraction: if 600 families did not pay, then the remaining families in the total of 3000 must have paid. This calculation is:

• Total number of families: 3000
• Families not paying tax: 600
• Successful outcomes (families paying tax): 3000 - 600 = 2400

Recognizing these successful outcomes allows you to apply probability formulas effectively. It provides direct insight into the event whose likelihood you are interested in determining.

The probability formula is a fundamental tool in evaluating how likely an event is to occur. In simple terms, the probability of an event is calculated by dividing the number of successful outcomes by the total number of possible outcomes in the sample space.
The formula is expressed as:

• Probability of Event (E) = \( \frac{\text{Number of Successful Outcomes}}{\text{Total Number of Outcomes}} \)

Applying this formula to our exercise:

• Number of successful outcomes: 2400
• Total number of outcomes: 3000
• Probability that a randomly selected family paid income tax = \( \frac{2400}{3000} = 0.8 \)

This probability value, 0.8, means that there is an 80% chance that a randomly chosen family from the apartment complex paid their income tax last year. By understanding and applying the probability formula, you can quantify the uncertainty of various events, which is invaluable across many fields of study and practical applications.