Problem 27 - Discrete Mathematics with Applications

Question

A survey of 475 customers at Chestnut Restaurant shows that of the three ice cream flavors - chocolate, strawberry, and vanilla - 65 like only chocolate, 75 like only strawberry, 85 like only vanilla, 100 like chocolate but not strawberry, 120 like strawberry but not vanilla, 140 like vanilla but not chocolate, and 65 like none of the flavors. A customer is selected at random from the survey. Find the probability that he likes: Chocolate.

Step-by-Step Solution

Answer

The probability that a randomly selected customer likes chocolate is \( \frac {4}{19} \).

Step 1: Identify relevant information from the survey

From the survey, we are provided the following information: - 65 customers like only chocolate - 100 customers like chocolate but not strawberry - 140 customers like vanilla but not chocolate - 65 customers like none of the flavors The other information provided is not directly related to the customers who like chocolate.

Step 2: Determine the total number of customers who like chocolate

The 100 customers who like chocolate but not strawberry includes the 65 customers who like only chocolate, as they also don't like strawberry. Therefore, we can determine the total number of chocolate likers by simply considering the 100 customers who like chocolate but not strawberry, as this already covers all the customers who like solely chocolate. So, we have a total of 100 customers who like chocolate.

Step 3: Calculate the probability of selecting a chocolate liker at random

Now, to find the probability that a randomly selected customer likes chocolate, we will divide the total number of customers who like chocolate by the total number of customers surveyed. Probability (customer likes chocolate) = Number of chocolate likers / Total number of customers surveyed Probability (customer likes chocolate) = \( \frac {100}{475} \)

Step 4: Simplify the probability

To simplify the probability, divide the numerator and denominator by their greatest common divisor (GCD). The GCD of 100 and 475 is 25. Simplified probability (customer likes chocolate) = \( \frac {100 \div 25}{475 \div 25} \) Simplified probability (customer likes chocolate) = \( \frac {4}{19} \) The probability that a randomly selected customer likes chocolate is \( \frac {4}{19} \).

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the collection of objects, known as sets. In the context of our problem, each preference category of ice cream flavors can be seen as a set. For example:

• The set of people who like only chocolate.
• The set of people who like chocolate but not strawberry.
• The set of people who like none of the flavors.

Each of these sets encompasses a group of customers with similar tastes. In set theory, we often look for intersections and unions of these sets.

The intersection represents members that are common between different sets, while the union represents members in any set. Understanding and manipulating these sets allows us to determine the number of people who share specific preferences.

Survey Analysis

Analyzing surveys effectively helps us draw conclusions based on data collected from specific groups. For the ice cream survey at Chestnut Restaurant, the key information gathered from participants was their ice cream flavor preferences.

From such data, we derive statistics such as:

• The number of customers liking only one flavor.
• The number preferring combinations of flavors.
• The number disliking all flavors.

Survey analysis involves organizing this data logically. For example, differentiating between those who like 'chocolate but not strawberry' and those who like 'only chocolate'. This aids in calculating probabilities or making other statistical inferences about the data collected, offering insights into consumer behavior.

Fractions

Fractions are a way to represent parts of a whole and are crucial in expressing probabilities. In our exercise, the probability that a randomly selected customer from the survey likes chocolate is expressed as a fraction.

• The numerator represents the number of favorable outcomes, which are the chocolate likers.
• The denominator stands for the total number of surveyed customers.

Calculating the probability involves dividing the number of chocolate likers by the total number of participants, forming the fraction: \( \frac {100}{475} \).

Understanding fractions as ratios helps in visualizing how one part relates to the whole. Simplifying these fractions into their simplest form shows the most reduced version of this ratio, making it easy to interpret the likelihood of the event.

GCD (Greatest Common Divisor)

The Greatest Common Divisor (GCD) is the largest number that divides exactly into two or more numbers. It plays a vital role in simplifying fractions. When we calculated the probability of a customer liking chocolate, we ended up with the fraction \( \frac{100}{475} \).

To simplify this fraction, we found the GCD of 100 and 475. The GCD in this context was 25. By dividing both the numerator and the denominator by this GCD, we obtained \( \frac{4}{19} \), a simpler equivalent of the original fraction.

• This simplification process helps in better understanding the proportion relationships.
• It makes the fraction easier to work with in further calculations or interpretations.

Hence, knowing about the GCD is crucial when dealing with probabilities in fraction form, as it leads to a cleaner, simplified version of the data.