Understanding Probability: The Likelihood of Multiples of 3 and 2 Between 1 and 20

When working with probability, it's often useful to explore specific scenarios to understand the underlying principles. In this article, we will delve into the probability of obtaining a multiple of both 3 and 2 from the numbers 1 to 20. This exploration will provide a clear understanding of how to approach such problems step-by-step and apply the concept of least common multiple (LCM).

Step-by-Step Approach to Finding Probability

1. Identify the Multiples of 6:
- The least common multiple (LCM) of 3 and 2 is 6. Therefore, we need to find all the multiples of 6 within the range of 1 to 20.

2. Count the Total Outcomes:
- The range of numbers from 1 to 20 includes a total of 20 integers. Therefore, our total number of outcomes is 20.

3. Count the Favorable Outcomes:
- The favorable outcomes are the multiples of 6 that lie within the specified range, which are 6, 12, and 18. Hence, the number of favorable outcomes is 3.

4. Calculate the Probability:
- The probability ( P ) of an event is given by the formula: [ P frac{text{Number of favorable outcomes}}{text{Total number of outcomes}} ] Applying this to our scenario: [ P frac{3}{20} ] In decimal form, this is equivalent to 0.15.

Conclusion

The probability of selecting a number that is a multiple of both 3 and 2 (i.e., a multiple of 6) from the numbers 1 to 20 is ( frac{3}{20} ) or 0.15. This calculation involves identifying the least common multiple (LCM), counting the favorable outcomes, and then applying the probability formula.

Knowing how to approach such problems is not only useful in mathematics but also in various real-world applications, such as in data analysis, statistics, and decision-making processes in business and research.