4. Probability
Multiplication Rule: Independent Events
2:50 minutesProblem 3.4.47
Textbook Question
Finding New Music In Exercises 45–48, use the pie chart, which shows the results of a survey of 513 music listeners who were asked about their primary source for new music. (Source: The Sound of AI)
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47. You choose nine music listeners at random. What is the probability that none of them say their primary source for new music is friends or social media?
Verified step by step guidance1
Step 1: Identify the relevant data from the problem. The pie chart provided does not pertain to the problem about music listeners, so we will focus on the textual information given in the problem. The survey involves 513 music listeners, and we are tasked with finding the probability that none of the nine randomly chosen listeners say their primary source for new music is friends or social media.
Step 2: Determine the probability that a single listener does not choose friends or social media as their primary source for new music. This requires knowing the proportion of listeners who chose friends or social media from the survey data. If this proportion is not explicitly provided, it must be calculated or assumed based on the problem context.
Step 3: Calculate the probability that a single listener does not choose friends or social media. If the proportion of listeners who chose friends or social media is p, then the probability of not choosing friends or social media is (1 - p).
Step 4: Use the binomial probability formula to calculate the probability that none of the nine listeners choose friends or social media. The formula for the probability of k successes in n trials is P(X = k) = (n choose k) * (p^k) * ((1-p)^(n-k)). Here, k = 0 (no successes), n = 9, and the probability of success (choosing friends or social media) is p.
Step 5: Simplify the formula for k = 0. The term (n choose k) becomes 1 when k = 0, and the formula simplifies to P(X = 0) = ((1-p)^n). Substitute the values of p and n into this formula to find the probability.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Probability
Probability is a measure of the likelihood that a particular event will occur, expressed as a number between 0 and 1. In this context, it involves calculating the chance that none of the selected music listeners cite friends or social media as their primary source for new music. Understanding how to compute probabilities, especially in scenarios involving multiple independent events, is crucial for solving the question.
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Independent Events
Independent events are those whose outcomes do not affect each other. In this scenario, the selection of each music listener is independent, meaning the choice of one listener does not influence the choices of others. This concept is essential for calculating the overall probability of multiple listeners not selecting a specific source, as it allows for the multiplication of individual probabilities.
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Complement Rule
The complement rule in probability states that the probability of an event not occurring is equal to one minus the probability of the event occurring. In this case, to find the probability that none of the nine listeners cite friends or social media, one can first determine the probability that at least one does and then subtract that from one. This approach simplifies the calculation and enhances understanding of the overall probability distribution.
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