Problem 13 - 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 vanilla, given that she does not like chocolate.

Step-by-Step Solution

Answer

The probability that a customer selected at random from the survey likes vanilla, given that she does not like chocolate, is \(\frac{28}{45}\).

Step 1: Calculate Joint Probability (A and B)

To calculate the joint probability of a customer liking vanilla and not liking chocolate, we will use the given data: 140 customers like vanilla but not chocolate. So, there are 140 customers who meet our criteria in the survey. \(P(A \cap B) = \frac{140}{475}\)

Step 2: Calculate Probability of Event B (Not Liking Chocolate)

To calculate the probability of a customer not liking chocolate, we will use the data provided on customers who like only strawberry, like only vanilla, and like none of the flavors, as all of these do not like chocolate. Number of people who don't like chocolate = (75 people who like only strawberry) + (85 people who like only vanilla) + (65 people who like none of the flavors) = 225 Thus, the probability of not liking chocolate: \(P(B) = \frac{225}{475}\)

Step 3: Calculate Conditional Probability (A|B)

Now, we will apply the conditional probability formula to calculate the probability of a customer liking vanilla, given that they don't like chocolate. \(P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{\frac{140}{475}}{\frac{225}{475}}\) Since the 475 in both numerators and denominators cancel out: \(P(A|B) = \frac{140}{225}\)

Step 4: Simplify the Probability

We can simplify the fraction to make the probability easier to understand: \(P(A|B) = \frac{140}{225} = \frac{28}{45}\) Therefore, the probability that a customer selected at random from the survey likes vanilla, given that she does not like chocolate, is \(\frac{28}{45}\).

Key Concepts

Joint Probability

Joint probability represents the likelihood that two events will occur simultaneously. In the context of the Chestnut Restaurant survey, we want to find the joint probability of customers liking vanilla while not liking chocolate. This requires identifying the subset of survey participants who meet these criteria.

From the survey data, it is stated that 140 customers like vanilla but not chocolate. These customers specifically meet the criteria of the joint event we are evaluating. Therefore, to determine the joint probability, we calculate as follows:

• The number of customers meeting our joint criteria: 140
• Total number of customers surveyed: 475

The formula for joint probability, represented as \(P(A \cap B)\), is calculated by dividing the number of customers who like vanilla and not chocolate by the total number of survey respondents:

\(P(A \cap B) = \frac{140}{475}\).

This fraction provides us with the joint probability of liking vanilla while not liking chocolate.

Probability Calculation

Calculating probability involves determining the likelihood of a specific event occurring out of the total number of possible outcomes. In this exercise, we want to find the probability of unique occurrences related to customer preferences.

To calculate the probability of a customer not liking chocolate, for example, we need to sum the numbers of each group who does not consume chocolate. These groups include those who like only strawberry, those who like only vanilla, and those who do not care for any of the flavors. Here's the breakdown:

• 75 customers like only strawberry.
• 85 customers like only vanilla.
• 65 customers are indifferent to all flavors.

By adding these numbers, we determine that 225 customers do not like chocolate.

Using the probability formula, the probability of a customer not liking chocolate \(P(B)\) is calculated as:

\(P(B) = \frac{225}{475}\).

This fraction tells us the likelihood that a randomly chosen customer does not favor chocolate.

Survey Analysis

Survey analysis involves interpreting collected data to answer specific questions or validate hypotheses. In this case, we are keen on discovering customer preferences for ice cream flavors at the Chestnut Restaurant, focusing on conditional probability.

Conditional probability helps us determine how the occurrence of one event affects another. Here, we're interested in finding out the probability that a customer likes vanilla, provided they do not like chocolate.

Using the conditional probability formula, we can reference the derived probabilities for event intersections and individual events. The formula is \(P(A|B) = \frac{P(A \cap B)}{P(B)}\), where:

• \(P(A \cap B)\) is the joint probability of liking vanilla and not liking chocolate.
• \(P(B)\) is the probability of not liking chocolate.

After finding \(P(A \cap B) = \frac{140}{475}\) and \(P(B) = \frac{225}{475}\), we plug these values into the formula:

\(P(A|B) = \frac{\frac{140}{475}}{\frac{225}{475}}\).

Simplifying this, the 475 in the numerators and denominators cancels, resulting in:

\(P(A|B) = \frac{140}{225}\).

Further simplification gives us \(P(A|B) = \frac{28}{45}\), representing the probability that a customer who doesn't like chocolate will indeed prefer vanilla.