Problem 14 - 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 customers 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. Find the probability that a customer selected at random from the survey: Likes chocolate, given that she does not like strawberry or vanilla.

Step-by-Step Solution

Answer

The probability that a customer selected at random from the survey likes chocolate, given that she does not like strawberry or vanilla, is \(0.5\) or \(50\%\).

Step 1: 1. Set up notations and understand the problem

Let A, B, and C denote events where customers like chocolate, strawberry, and vanilla respectively. We are given the number of customers who like each flavor, and we want to find the probability of liking chocolate (event A) given not liking strawberry (event B') and not liking vanilla (event C'). We can represent this probability as P(A|B'C').

Step 2: 2. Find the number of customers who like only chocolate, strawberry and vanilla

We are given that 65 customers like only chocolate, 75 like only strawberry, and 85 like only vanilla. We use this information to determine the number of customers in each event A, B, and C. Number of customers who like only chocolate (A): 65 Number of customers who like only strawberry (B): 75 Number of customers who like only vanilla (C): 85

Step 3: 3. Find the number of customers who don't like strawberry or vanilla

We want to find the number of customers who don't like strawberry (B') and don't like vanilla (C'). We are given that 65 customers like none of the flavors. So, the number of customers who don't like strawberry or vanilla is equal to the number of customers who like only chocolate plus the customers who don't like any flavor: Number of customers who don't like strawberry or vanilla (B'C'): 65 (like only chocolate) + 65 (like none of the flavors) = 130

Step 4: 4. Calculate the conditional probability

Now, we need to calculate P(A|B'C'). The conditional probability can be calculated using the formula: P(A|B'C') = P(A ∩ B'C') / P(B'C') The probability P(A ∩ B'C') corresponds to the number of customers who like only chocolate since these customers like chocolate and don't like strawberry or vanilla. So, P(A ∩ B'C') = 65 customers. The probability P(B'C') is the total number of customers not liking strawberry/vanilla, which we found to be 130 customers in step 3. P(A|B'C') = P(A ∩ B'C') / P(B'C') = 65 / 130 = 1/2 = 0.5 So, the probability that a customer selected at random from the survey likes chocolate, given that she does not like strawberry or vanilla, is 0.5 or 50%.

Key Concepts

Set Theory

Set theory is all about understanding and working with groups or collections of items, which are often referred to as 'sets'. In the context of probability and surveys, these sets can represent customers with specific preferences. For our ice cream scenario, each set represents customers who like a particular flavor of ice cream, such as chocolate, strawberry, or vanilla.
Consider the sets:

• Set A: Customers who like chocolate,
• Set B: Customers who like strawberry,
• Set C: Customers who like vanilla.

In problems like this, we often combine and contrast these sets to find interesting results, such as who likes more than one flavor or who doesn't like a certain flavor.
Using information such as '65 customers like only chocolate,' we're identifying a subgroup within Set A that does not overlap with any other sets. Set theory allows us to categorize and quantify such relationships, making it a powerful tool in probability and data analysis.

Probability Theory

Probability theory is the mathematical framework for analyzing and predicting the likelihood of events. In many cases, like in our exercise, we focus on conditional probabilities. This means determining the probability of an event occurring under a specific condition.
Here, we want to find the probability of a random customer liking chocolate, given they do not like strawberry or vanilla. Our random events are:

• Event A: The customer likes chocolate,
• Event B': The customer does not like strawberry,
• Event C': The customer does not like vanilla.

The formula for conditional probability is:\[ P(A|B'C') = \frac{P(A \cap B'C')}{P(B'C')} \]Where:

• \(P(A \cap B'C')\) is the probability that both event A happens and conditions B' and C' are met,
• \(P(B'C')\) is the total probability of the condition where strawberry and vanilla are not liked.

Understanding how to navigate these probabilities is crucial for accurately interpreting survey and experimental data.

Venn Diagram

A Venn diagram is a diagrammatic way to visually represent sets and their relationships with each other. It’s an invaluable tool in set theory and probability, especially in conditional probability scenarios.
With Venn diagrams, each circle represents a set. Overlapping areas show where the sets intersect, meaning elements that belong to multiple sets.
In our ice cream preference survey:

• One circle could represent customers who like chocolate,
• Another circle would show those who like strawberry,
• The last would contain those who like vanilla.

Using Venn diagrams, we visualize complex relationships and easily identify overlapping preferences and exclusive choices, like those who only like one flavor or none.
This makes Venn diagrams a simple yet powerful visual aid for solving problems in probability and set theory, allowing us to better understand how different events and conditions relate to one another.