Probability Calculation in a Randomly Generated Integer Range

Understanding the concepts of probability, conditional probability, and even numbers can be quite fascinating, especially when dealing with a specific range of integers. In this article, we explore the probability that a randomly generated number between 1 and 66 is less than 25, given that the number is even. This involves a step-by-step analysis to determine the correct probability.

Problem Statement and Assumptions

First, let's clarify the problem with some assumptions. When you mention a number, it is reasonable to assume that we are dealing with integers. We are to find the probability that a randomly generated integer, ( n ), is less than 25 given that it is even, where ( n ) is between 1 and 66.

Defining Sets and Using Probability Concepts

We define two sets to frame our problem:

Set ( A ): This set includes all even positive integers between 1 and 66. Set ( B ): This set includes all integers between 1 and 25.

What we need to find is the conditional probability of ( B ) given ( A ), denoted by ( P(B | A) ). Using the formula for conditional probability, we can express ( P(B | A) ) as:

[ P(B | A) frac{P(B cap A)}{P(A)} ]

Calculating ( P(B cap A) )

The intersection ( B cap A ) refers to all even positive integers between 1 and 25. Since the even numbers between 1 and 25 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24, there are 12 such integers. Therefore, the probability ( P(B cap A) ) is:

[ P(B cap A) frac{12}{66} ]

Calculating ( P(A) )

The total number of even positive integers between 1 and 66 is 33 (i.e., 2, 4, 6, ..., 66). Hence, the probability ( P(A) ) is:

[ P(A) frac{33}{66} frac{1}{2} ]

Calculating the Conditional Probability ( P(B | A) )

Now, substituting the values in the conditional probability formula:

[ P(B | A) frac{frac{12}{66}}{frac{33}{66}} frac{12}{33} frac{4}{11} ]

Alternative Approach

Alternatively, we can also approach this using a simpler counting method. Since the range of numbers we are dealing with is 1 to 66, the total number of possible outcomes is 66. The even numbers in this range are 2, 4, 6, ..., 66. This means there are 33 even numbers (determined without proof that there are 33 even integers between 1 and 66).

Now, the number of even numbers less than 25 is limited to 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, and 24, which is 12 in total. Therefore, the probability is:

[ P(B | A) frac{12}{33} frac{4}{11} ]

Conclusion

The conditional probability that a randomly generated even integer between 1 and 66 is less than 25 is ( frac{4}{11} ). This approach showcases the elegance of probability theory and the importance of clear set definitions and the application of fundamental probability formulas.

Key Points:

The intersection of two sets determines the elements that satisfy both conditions. The total number of outcomes must be considered carefully. Conditional probability formulas provide a clear and systematic way to solve such problems.