Understanding the Probability of Divisibility

In the field of probability and statistics, determining the likelihood of a particular event is essential. This article focuses on calculating the probability that a number selected at random between 1 and 100 is divisible by either 2 or 7. We will employ the principle of inclusion-exclusion to achieve this.

The Principle of Inclusion-Exclusion

The principle of inclusion-exclusion is a fundamental concept in combinatorics and probability. It provides a method to calculate the size of the union of multiple sets that may overlap. This principle is particularly useful when dealing with problems involving multiple conditions.

Step-by-Step Calculation

Total Numbers

First, we need to determine the total number of integers between 1 and 100, inclusive. There are a total of 100 numbers in this range.

Numbers Divisible by 2

An arithmetic sequence can be used to identify the numbers divisible by 2. The sequence starts at 2 and ends at 100 with a common difference of 2.

The formula for the number of terms in an arithmetic sequence is given by:

n frac{l - a}{d} 1

Substituting the values:

n frac{100 - 2}{2} 1 50

Thus, there are 50 numbers divisible by 2.

Numbers Divisible by 7

Similarly, the numbers divisible by 7 form another arithmetic sequence: 7, 14, 21, ..., 98. Here, a 7, l 98, and d 7.

Using the formula:

n frac{98 - 7}{7} 1 14

Therefore, there are 14 numbers divisible by 7.

Numbers Divisible by Both 2 and 7 (i.e., Divisible by 14)

The numbers divisible by both 2 and 7 are also divisible by their least common multiple (LCM), which is 14. The sequence is: 14, 28, 42, ..., 98. Here, a 14, l 98, and d 14.

Using the formula:

n frac{98 - 14}{14} 1 7

Hence, there are 7 numbers divisible by 14.

Applying Inclusion-Exclusion Principle

According to the principle of inclusion-exclusion:

Number of numbers divisible by either 2 or 7 (Number of numbers divisible by 2) (Number of numbers divisible by 7) - (Number of numbers divisible by both 2 and 7)

Substituting the values:

50 14 - 7 57

Therefore, there are 57 numbers between 1 and 100 that are divisible by either 2 or 7.

Calculating the Probability

The probability that a randomly chosen number between 1 and 100 is divisible by either 2 or 7 is given by:

P frac{57}{100} 0.57

This probability can be expressed as 57% or 0.57.

In conclusion, the probability that a number chosen at random between 1 and 100 is divisible by either 2 or 7 is 0.57 or 57%.