Question

A national survey determined the breakfast beverage of choice for American high school students. The results are summarized in the table above. Based on this information, which of the following is closest to the probability that a student drinks coffee, given that she does not drink a cold drink at breakfast? A) $0.84$ B) $0.66$ C) $0.32$ D) $0.27$

Short answer

The probability that a student drinks coffee, given that she does not drink a cold drink at breakfast, is approximately $0.66$. So, the correct answer is option B.

Step-by-step solution

Understand the problem and given information

A national survey determined the breakfast beverage of choice for American high school students. Based on this information, we have to find the probability that a student drinks coffee, given that she does not drink a cold drink at breakfast.

Recall the formula for conditional probability

The formula for conditional probability is: $P(A|B)=\frac{P(A\cap B)}{P(B)}$ Where: - $P(A|B)$ is the probability of event A occurring, given that event B occurs. - $P(A\cap B)$ is the probability of both A and B occurring. - $P(B)$ is the probability of event B occurring. In this case: - A is the event that a student drinks coffee. - B is the event that a student does not drink a cold drink at breakfast.

Calculate the relevant probabilities

First, let's find the probabilities of event A and event B. 1. To find the probability of event A (a student drinks coffee), we need to add the probability of students drinking coffee, given that they drink a cold drink, and the probability of students drinking coffee, given that they do not drink a cold drink. We don't have the exact value ​​in the given table but we will use the values to find it. 2. To find the probability of event B (student does not drink a cold drink at breakfast), we need to check the second row of the given table. Add the probability of students drinking each selection given they do not drink a cold drink.

Apply the conditional probability formula

Now that we have the required probabilities, we can apply the formula for conditional probability. $P(\text{Coffee}|\text{Not Cold})=\frac{P(\text{Coffee}\cap \text{Not Cold})}{P(\text{Not Cold})}$ Then, we can plug in the values we found in the previous step and calculate the conditional probability.

Compare the results with the given options

After calculating the conditional probability, compare the result with the given options (A, B, C, and D) and choose the one that is closest to the answer. The correct answer will be the option closest to the conditional probability we found.

Key concepts

Probability Concepts

Probability is all about measuring the chance or likelihood of an event happening. In terms of numbers, it ranges from 0 to 1. A probability of 0 means something will not happen, while a probability of 1 means it will absolutely happen. In the context of conditional probability, we focus on the probability of an event occurring under a specific condition. For example, in our exercise, we are interested in the likelihood of a student drinking coffee given that they do not consume a cold drink at breakfast. Conditional probability uses the formula: $$P(A|B)=\frac{P(A\cap B)}{P(B)}$$ This equation helps us understand the probability of event A happening when event B is already known to occur. This is powerful because it narrows down our set of possibilities, making our prediction more precise. In surveys, understanding conditional probabilities can help identify relationships between different lifestyle choices or preferences.

High School Statistics

Statistics is a branch of mathematics that focuses on gathering, analyzing, and interpreting data. In high school statistics, students deal with various types of data and learn concepts like mean, median, mode, and probability. Probability is a central part of statistics since it provides a way to anticipate the behavior of data based on rules and formulas. For high school students, mastering the concept of probability, including conditional probability, is crucial since it sets the foundation for more advanced statistical studies in college. These skills are particularly useful when analyzing the outcomes of surveys or experiments. By calculating probabilities, students can make informed predictions and decisions based on data. This knowledge empowers students to apply statistical techniques to real-world problems, where understanding the likelihood of different events is key to sound decision-making.

Survey Analysis

Survey analysis involves collecting and interpreting data from surveys. Surveys are a popular method to gather information from a specific group of people on various topics like preferences, habits, or opinions. In the exercise, we're dealing with a national survey on breakfast beverage choices among high school students. Analyzing data from surveys includes determining probabilities. This can provide insights into patterns and relationships, such as the popularity of coffee among students who prefer hot beverages over cold ones. Survey analysis often requires using conditional probabilities to discern how one subset behaves different from another. By applying concepts like conditional probability, we can not only find answers to our questions but also draw deeper insights from our data. This ability to analyze survey data helps in making predictions or decisions based on the past habits and choices of respondents.