Probability Analysis: Divisibility by 4 but Not by 8 within the Range 1 to 100

Understanding the probability of a number being divisible by 4 but not by 8 within a specific range can be quite instructive. In this article, we will delve into the steps needed to solve such a problem and explore the broader implications of divisibility within the range from 1 to 100. This analysis incorporates probability, arithmetic sequences, and practical applications of number theory.

Introduction to the Problem

The problem at hand involves finding the probability that a randomly selected number from 1 to 100 is divisible by 4 but not by 8. Let's break down the problem step by step and use mathematical reasoning to arrive at the solution.

Step-by-step Solution

Step 1: Count the Total Numbers

There are 100 numbers in the range from 1 to 100.

Step 2: Count the Numbers Divisible by 4

The numbers divisible by 4 within the range of 1 to 100 form an arithmetic sequence: 4, 8, 12, ..., 100.

We can use the formula for the n-th term of an arithmetic sequence:

l a (n-1)d

Where:

l last term (100) a first term (4) d common difference (4)

Solving for n:

100 4 (n-1)4

100 - 4 (n-1)4

96 (n-1)4

24 (n-1)

n 24 1

n 25

So, there are 25 numbers divisible by 4.

Step 3: Count the Numbers Divisible by 8

The numbers divisible by 8 within the range of 1 to 100 form another arithmetic sequence: 8, 16, 24, ..., 96.

Using the same formula for the n-th term of an arithmetic sequence:

l a (n-1)d

Where:

l last term (96) a first term (8) d common difference (8)

Solving for n:

96 8 (n-1)8

96 - 8 (n-1)8

88 (n-1)8

11 (n-1)

n 12

So, there are 12 numbers divisible by 8.

Step 4: Count the Numbers Divisible by 4 but Not by 8

To find the numbers that are divisible by 4 but not by 8, subtract the count of numbers divisible by 8 from those divisible by 4:

Count 25 - 12 13

So, there are 13 numbers divisible by 4 but not by 8.

Step 5: Calculate the Probability

The probability P that a randomly picked number from 1 to 100 is divisible by 4 but not by 8 is given by the ratio of the favorable outcomes to the total outcomes:

P frac{13}{100}

Thus, the probability is frac{13}{100} or 0.13.

Alternative Explanation

Alternatively, you might consider that half of the numbers divisible by 4 are also divisible by 8. Therefore, the probability of picking a number that is divisible by 4 but not by 8 is 0.125. However, to avoid rounding issues and ensure precision, the precise count method is preferred.

Conclusion

In conclusion, the probability that a number picked from 1 to 100 is divisible by 4 but not by 8 is frac{13}{100} or 0.13. This analysis provides a clear and detailed step-by-step approach to solving similar problems involving divisibility and probability.

For further exploration, you can apply similar methods to other ranges and numbers, or extend the problem to include more complex divisibility conditions. Understanding these concepts can be crucial in various fields, including mathematics, statistics, and everyday problem-solving.