Problem 26 - 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: All flavors.

Step-by-Step Solution

Answer

The probability that a randomly chosen customer likes all three flavors of ice cream is approximately 13.7%.

Step 1: Analyze the data provided

The survey provides information on the number of customers who like different combinations of ice cream flavors. The data tells us the following: - 65 customers like only chocolate - 75 customers like only strawberry - 85 customers like only vanilla - 100 customers like chocolate but not strawberry - 120 customers like strawberry but not vanilla - 140 customers like vanilla but not chocolate - 65 customers like none of the flavors

Step 2: Find the number of customers who like each pair of flavors

From the information provided, we can calculate the number of people who like each pair of flavors. - 100 customers like chocolate but not strawberry, so the number of customers who like both chocolate and strawberry is (65 + 100) - 65 = 100. - 120 customers like strawberry but not vanilla, so the number of customers who like both strawberry and vanilla is (75 + 120) - 75 = 120. - 140 customers like vanilla but not chocolate, so the number of customers who like both chocolate and vanilla is (85 + 140) - 85 = 140.

Step 3: Find the number of customers who like all three flavors

We can use the information from Step 2 to calculate the number of customers who like all three flavors. - There are 100 customers who like both chocolate and strawberry. These customers are included in both the "100 customers who like chocolate but not strawberry" and the "120 customers who like strawberry but not vanilla" groups. Therefore, the number of customers who like all three flavors is 100 - (100 - 65) = 65.

Step 4: Calculate the probability that a randomly chosen customer likes all three flavors

Now that we know the number of customers who like all three flavors, we can calculate the probability that a random customer likes all three flavors. The probability is equal to the number of customers who like all three flavors divided by the total number of customers. Probability = (number of customers who like all flavors) / (total number of customers) = 65 / 475 ≈ 0.137 So, the probability that a randomly chosen customer likes all three flavors of ice cream is approximately 13.7%.

Key Concepts

Set Theory

Set theory is a branch of mathematical logic that deals with the collection of objects, known as sets, which are considered as wholes. These objects could be anything: numbers, symbols, or even ice cream flavors, as seen in the given exercise. When we refer to people liking only one flavor, we’re essentially talking about distinct sets. For instance, the set of customers who like only chocolate denotes a unique group with no overlap with those who prefer just strawberry or vanilla.

In applying set theory to solve the given exercise, we are to figure out how these sets intersect — like determining how many customers belong to the set of all flavors, representing those who like chocolate, strawberry, and vanilla simultaneously. Set theory provides the foundational language and rules for dealing with these groups and their combinations in an organized manner, paving the way to analyze data and calculate probabilities within any given population.

Understanding these basic concepts is essential for dissecting complex problems, such as those we face in the world of probability, which intimately relies on set theory.

Venn Diagrams

Venn diagrams are a visual representation of set theory, used to illustrate the relationships between different sets. These diagrams often consist of overlapping circles, each representing a set. Where the circles overlap, we find the intersection of the sets, or objects that are members of both (or all) sets involved.

For our ice cream flavor problem, a Venn diagram would be handy to depict the intricate relationships between those who like chocolate, strawberry, and vanilla. It's much easier to visualize the data provided - we can clearly see the separate sets of flavor likers, the intersections for those who like two flavors, and the central overlapping region for those who adore all three. It also gives us a clear view of the people who don't like any flavors at all. Presenting the information this way can significantly improve comprehension and provide a more intuitive route to the probability calculations needed to solve the exercise.

Probability Calculation

Probability calculation involves quantifying the likelihood of a particular event occurring. It’s typically expressed as a fraction between 0 and 1, with 0 indicating impossibility and 1 denoting certainty. In discrete mathematics, and particularly in problems concerning surveys, such as the ice cream flavors preference, we use probability to deduce how likely it is for a random event, like selecting a customer with a certain preference, to occur.

To find the probability of a customer liking all three flavors, as the exercise poses, we count the number of customers who fall into the 'all flavors' category and divide it by the total number of customers surveyed. This simplistic yet powerful approach gives us the proportion of the whole that meets our criteria — a critical step in understanding how likely certain outcomes are in various scenarios.

By mastering probability calculation, students can handle real-world problems with confidence, whether it's in predicting outcomes or making informed decisions based on statistical data.