Probability of Selecting a Two-Digit Multiple of 5: A Comprehensive Guide

Determining the probability of selecting a two-digit number that is a multiple of 5 involves a series of logical steps and mathematical calculations. This guide will walk you through the process, ensuring you understand the underlying principles and methods involved.

Introduction

The probability of selecting a number with specific characteristics, such as being a multiple of 5, in a given range can be calculated using fundamental concepts in probability theory. Specifically, we will focus on the range of two-digit numbers and the multiples of 5 within that range.

Identifying the Range of Two-Digit Numbers

The range of two-digit numbers is straightforward. A two-digit number falls between 10 and 99, inclusive. Therefore, we can consider any number within this range to be a valid candidate for our analysis.

Counting the Total Number of Two-Digit Numbers

To find the total number of two-digit numbers, we can use the following calculation:

( text{Total number of two-digit numbers} 99 - 10 1 90 )

This formula accounts for all the numbers from 10 to 99, inclusive, giving us a total of 90 two-digit numbers.

Identifying the Two-Digit Multiples of 5

Numbers that are multiples of 5 end in either 0 or 5. This property holds true for all numbers, whether they have 2, 500, or any other number of digits. In our case, we are interested in the two-digit multiples of 5 within the range of 10 to 99.

The smallest two-digit multiple of 5 within this range is 10, and the largest is 95. We can list these multiples as follows:

10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60, 65, 70, 75, 80, 85, 90, 95

Counting the Number of Terms in the Sequence

The sequence of two-digit multiples of 5 (10, 15, 20, ..., 95) forms an arithmetic sequence. An arithmetic sequence is defined by the formula:

( a_n a_1 (n-1)times d )

Where:

( a_1 10 ) (the first term) ( d 5 ) (the common difference) ( a_n 95 ) (the last term)

To find the number of terms in the sequence, we use the formula:

( a_n a_1 (n-1)times d )

Setting ( a_n 95 ), we can solve for ( n ):

( 95 10 (n-1)times 5 )

( 95 - 10 (n-1)times 5 )

( 85 (n-1)times 5 )

( n - 1 frac{85}{5} 17 )

( n 18 )

This means there are 18 two-digit multiples of 5.

Calculating the Probability

The probability of selecting a two-digit number that is a multiple of 5 is given by the ratio of the number of favorable outcomes (multiples of 5) to the total number of outcomes (two-digit numbers):

( P frac{text{Number of two-digit multiples of 5}}{text{Total number of two-digit numbers}} frac{18}{90} frac{1}{5} )

This simplifies to:

( P 0.2 )

Conclusion

In conclusion, the probability of selecting a two-digit number that is a multiple of 5 is 0.2 or ( frac{1}{5} ). This result can be generalized to any range of numbers where the last digit is either 0 or 5.

Additional Insights

Numbers that are even multiples of 5 will always end in 0. For example, among the two-digit numbers, 10, 20, 30, 40, 50, 60, 70, 80, and 90 are even multiples of 5. Similarly, numbers that end in 5 but are not even are odd multiples of 5, such as 15, 25, 35, 45, 55, 65, 75, 85, and 95.

A key takeaway is that the probability of the last digit being 5 or 0, within a specific range of numbers, can be determined by the distribution of these digits. In this case, for a random selection, the probability is ( frac{1}{5} ).

Understanding these principles can be useful in various applications, including data analysis, statistics, and probability theory.