Problem 48 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Fifty raffle tickets are numbered 1 through 50 , and one of them is drawn at random. What is the probability that the number is a multiple of 5 or 7 ? Consider the following "solution": Since 10 tickets bear numbers that are multiples of 5 and since 7 tickets bear numbers that are multiples of 7 , we conclude that the required probability is $$ \frac{10}{50}+\frac{7}{50}=\frac{17}{50} $$ What is wrong with this argument? What is the correct answer?

Step-by-Step Solution

Answer

The given solution incorrectly adds the probabilities without accounting for the double-counting of the number 35, which is a multiple of both 5 and 7. The correct answer can be calculated using the Inclusion-Exclusion principle. There are 10 multiples of 5, 7 multiples of 7, and 1 common multiple (35) in the range of ticket numbers (1 to 50). Therefore, the total number of tickets with numbers that are multiples of 5 or 7 is 16, and the correct probability is \(\frac{16}{50} = \frac{8}{25}\).

Step 1: Find the multiples of 5 and 7 within 1 to 50

List out all multiples of 5 and 7 in the range of ticket numbers (1 to 50). Multiples of 5: \(5, 10, 15, 20, 25, 30, 35, 40, 45, 50\) (10 numbers) Multiples of 7: \(7, 14, 21, 28, 35, 42, 49\) (7 numbers)

Step 2: Identify common multiples

Find the common multiples of 5 and 7 within the ticket numbers range since these have been counted twice. Common multiples of 5 and 7: \(35\) (1 number)

Step 3: Apply the Inclusion-Exclusion principle

To compute the total number of tickets with numbers that are multiples of 5 or 7, subtract the number of common multiples from the sum of multiples of 5 and multiples of 7 to avoid double-counting: Total = Multiples of 5 + Multiples of 7 - Common multiples Total = 10 + 7 - 1 = 16

Step 4: Compute the probability

Now, we can calculate the probability by dividing the total number of tickets with numbers that are multiples of 5 or 7 by the total number of tickets: Probability = \(\frac{16}{50} = \frac{8}{25}\) So, the correct answer is the probability \(= \frac{8}{25}\).

Key Concepts

Inclusion-Exclusion Principle

In the world of probability and counting, the Inclusion-Exclusion Principle is a powerful tool. This principle helps us correctly count the number of elements in the union of multiple sets, by taking into account overlaps where elements might be double-counted.

When counting items such as raffle tickets with numbers that are multiples of 5 or 7, the principle becomes essential. Initially, you might think to simply add together the numbers of multiples of 5 and 7. However, this approach includes a common error: overcounting items present in both groups.

The critical insight provided by the Inclusion-Exclusion Principle is to subtract these common items that are counted twice. In our example, the number 35 is a multiple of both 5 and 7. If not correctly accounted for, it leads to an incorrect total, which further affects the probability calculation. Hence, the correct number of elements is the sum of different multiples, minus the overlap.

Multiples

Multiples are integers that can be divided by a specific number without leaving a remainder. In practical terms, they are the results of multiplying that number by integers. For instance, multiples of 5 are 5, 10, 15, and so on.

In our problem, we are focused on raffle tickets numbered 1 through 50. We need to calculate how many of these numbers are multiples of either 5 or 7. Finding these numbers is relatively straightforward:

• For 5: 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 (10 numbers)
• For 7: 7, 14, 21, 28, 35, 42, 49 (7 numbers)

Notice that 35 appears in both lists because it is a multiple of both 5 and 7.

Counting Problem

A Counting Problem involves determining how many elements satisfy certain conditions from a finite set. It's a fundamental part of probability and combinatorics. In our raffle ticket example, we face a classic counting problem: identifying tickets that are either multiples of 5 or 7.

Initially, it seems simple to count achievements in separate group totals. However, complications arise due to shared elements (common multiples). The problem-solving challenge lies in correctly categorizing and counting these shared elements only once.

Our solution involves systematically listing the multiples and correctly applying the Inclusion-Exclusion Principle to ensure each element is counted the appropriate number of times. This careful counting avoids common pitfalls like overcounting shared elements.

Mathematics

Mathematics forms the foundation for solving complex counting and probability problems. It provides the tools and principles that allow us to handle seemingly simple problems accurately.

In this exercise, mathematical reasoning brings us from a straightforward observation to a precise solution. By understanding multiples, employing combinatorial techniques like the Inclusion-Exclusion Principle, and wielding probability, we navigate from data misinterpretations to a clear answer.

Mathematics, in this context, is not just about numbers but understanding the relations and structures between them. It encourages a mindset of checking every assumption, verifying results, and ensuring each calculation leads to a meaningful conclusion. Through tools like latex and logical structuring, mathematics becomes a language of clarity and precision.