Problem 28 - 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: Exactly two flavors.

Step-by-Step Solution

Answer

The probability that a randomly chosen customer likes exactly two flavors is \( \dfrac{72}{95} \).

Step 1: Count Customers Who Like Exactly Two Flavors

Let's use the given data to find out how many people like exactly two flavors. - 100 like chocolate but not strawberry. Since this group doesn't like strawberry, they can only like chocolate and vanilla. - 120 like strawberry but not vanilla. Since this group doesn't like vanilla, they can only like chocolate and strawberry. - 140 like vanilla but not chocolate. Since this group doesn't like chocolate, they can only like strawberry and vanilla. So, by adding these three counts, we get the total number of customers who like exactly two flavors: \( 100 + 120 + 140 = 360 \)

Step 2: Calculate Probability

Now that we know 360 customers like exactly two flavors, we can find the probability that a randomly selected customer likes exactly two flavors. The probability is the ratio of the favorable outcome (360) to the total number of possible outcomes (475). So, the probability is: \( P(\text{Exactly two flavors}) = \dfrac{360}{475} \)

Step 3: Simplify the Probability

We can simplify the fraction to make it easier to understand: \( P(\text{Exactly two flavors}) = \dfrac{360}{475} = \dfrac{72}{95} \) So, the probability that a randomly chosen customer likes exactly two flavors is \( \dfrac{72}{95} \).

Key Concepts

Ice Cream Survey

Surveys are a common method in statistics for gathering information from a large group through a representative sample. In this ice cream survey, Chestnut Restaurant collected data from 475 customers about their ice cream preferences. Each customer could express a liking for chocolate, strawberry, and vanilla flavors.

Survey results are crucial for analyzing trends and understanding customer preferences. For example, the survey shows that:

• 65 customers like only chocolate.
• 75 customers like only strawberry.
• 85 customers like only vanilla.
• 100 customers like chocolate but not strawberry (and hence must also like vanilla).
• 120 customers like strawberry but not vanilla (and hence must also like chocolate).
• 140 customers like vanilla but not chocolate (and hence must also like strawberry).
• 65 customers like none of these flavors.

This breakdown shows that a significant number of customers have overlapping preferences. Understanding these overlaps is essential for businesses in tailoring their offerings to meet customer demand. An ice cream parlor, for instance, could adjust its inventory based on the popularity of these combinations.

Discrete Mathematics

Discrete Mathematics is a branch of mathematics that deals with distinct and separate values. It is prominently used in real-world applications like computer science, logistics, and data analysis. This field involves structures such as graphs, integers, and, as seen in our problem, set theory.

In the context of the ice cream survey, discrete mathematics helps to analyze sets. Each flavor preference (chocolate, strawberry, and vanilla) can be seen as a separate set of customers.

Set theory is fundamental in figuring out how many customers like exactly two flavors. By using concepts like intersection and differences of sets, we count groups that have shared elements. For example:

• The set of customers who like chocolate but not strawberry is a subset that intersect with vanilla lovers.
• You can similarly intersect other flavors to find double preferences without including all three flavors or those who like none.

Discrete mathematics thus provides tools for logical reasoning necessary to solve problems that involve finite choices, such as our survey analysis.

Two Flavors Probability

Probability theory is the study of uncertainty and helps us measure the likelihood of certain events happening. In this case, probability theory is used to determine how likely it is for a Chestnut Restaurant customer to like exactly two ice cream flavors.

The problem boils down to dividing the number of desired events (360 customers who like two flavors) by the total number of possible events (all 475 surveyed customers). This gives us:\( P(\text{Exactly two flavors}) = \frac{360}{475} \)
Simplifying the fraction helps in better understanding and communicating the probability:\( P(\text{Exactly two flavors}) = \frac{72}{95} \)
The probability of finding a customer with exactly two flavor preferences this way equates to approximately 0.7579, or about 75.8% when rounded up. This demonstrates a high likelihood, showing that most customers surveyed do indeed enjoy combinations of flavors, rather than single preferences or none at all.Being aware of such probabilities helps businesses better plan their inventory and marketing strategies to align with customer tastes.