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Ch. 3 - Probability
Larson - Elementary Statistics: Picturing the World 8th Edition
Larson8th EditionElementary Statistics: Picturing the WorldISBN: 9780137493470Not the one you use?Change textbook
All textbooksLarson 8th EditionCh. 3 - ProbabilityProblem 3.4.43a
Chapter 3, Problem 3.4.43a

Shuffle Play You use a shuffle playback feature to randomly play songs in a playlist. The playlist of 56 songs includes 15 instrumental songs.
a. What is the probability that the first three songs to play are instrumental songs? (Assume a song cannot be repeated.)

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Step 1: Understand the problem. We are tasked with finding the probability that the first three songs played from a playlist of 56 songs (15 of which are instrumental) are all instrumental. Since songs cannot be repeated, this is a problem involving dependent probabilities.
Step 2: Recall the formula for probability. The probability of multiple dependent events occurring is the product of their individual probabilities. For this problem, the probability of selecting an instrumental song changes after each selection because the total number of songs and the number of instrumental songs both decrease.
Step 3: Calculate the probability of the first song being instrumental. The probability is the ratio of the number of instrumental songs to the total number of songs: \( P(\text{First song instrumental}) = \frac{15}{56} \).
Step 4: Calculate the probability of the second song being instrumental, given that the first song was instrumental. After one instrumental song is played, there are 14 instrumental songs left and 55 songs in total: \( P(\text{Second song instrumental | First song instrumental}) = \frac{14}{55} \).
Step 5: Calculate the probability of the third song being instrumental, given that the first two songs were instrumental. After two instrumental songs are played, there are 13 instrumental songs left and 54 songs in total: \( P(\text{Third song instrumental | First two songs instrumental}) = \frac{13}{54} \). Multiply these probabilities together to find the overall probability: \( P(\text{All three songs instrumental}) = \frac{15}{56} \times \frac{14}{55} \times \frac{13}{54} \).

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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 quantifies the chance of selecting instrumental songs from a playlist. The probability of an event can be calculated by dividing the number of favorable outcomes by the total number of possible outcomes.
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Introduction to Probability

Combinatorial Counting

Combinatorial counting involves determining the number of ways to choose or arrange items from a set. In this scenario, it is essential to calculate how many ways the first three songs can be selected from the 15 instrumental songs out of the total 56. This concept helps in understanding how to compute probabilities in situations where order matters.
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Fundamental Counting Principle

Dependent Events

Dependent events are events where the outcome of one event affects the outcome of another. In this question, the selection of the first song influences the available choices for the second and third songs, as songs cannot be repeated. Understanding this concept is crucial for accurately calculating the probability of sequential events.
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Probability of Multiple Independent Events
Related Practice
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