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Chapter 5: Problem 14

Out of 100 songs on a playlist, 15 are of your favorite artist. You decide to randomly play one track from this playlist. a. State the sample space for the possible outcomes. b. State the probability for each possible outcome. c. What is the probability that the track chosen randomly from the playlist is of your favorite artist? d. What is the probability that the track chosen randomly from the playlist is not of your favorite artist?

Short Answer

Expert verified
a. 100 songs; b. Each song has a probability of \( \frac{1}{100} \); c. 0.15; d. 0.85.

Step by step solution

01

Define the Sample Space

The sample space for the possible outcomes is the total number of songs in the playlist. There are 100 songs, so the sample space is all these songs, labeled as Song 1, Song 2, ..., Song 100.
02

Probability for Each Outcome

Since each song has an equal chance of being played, the probability of playing any specific song is the same. The probability for each song is given by the formula: \( \frac{1}{\text{Total Number of Songs}} = \frac{1}{100} \).
03

Probability of Playing a Favorite Song

To find the probability of playing a song by your favorite artist, who has 15 songs on the playlist, use the formula:\[ \text{Probability} = \frac{\text{Number of favorite artist's songs}}{\text{Total number of songs}} = \frac{15}{100} = 0.15 \].
04

Probability of Playing a Non-favorite Song

The probability of playing a song that is not by your favorite artist can be calculated by subtracting the probability of playing a favorite artist's song from 1. Hence, \( \text{Probability} = 1 - 0.15 = 0.85 \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sample Space
When dealing with probability, the term "sample space" refers to all the possible outcomes in a random experiment. In the context of the playlist exercise, the sample space is the set of all songs available on the playlist.

A playlist with 100 songs forms a sample space of 100 distinct possibilities. You can think of each song as a single outcome labeled from Song 1 to Song 100. This method of labeling helps in visualizing and accounting for every possible track that could be randomly selected.

Having a clearly defined sample space is crucial because it lays the foundation for calculating probabilities. It ensures you are considering all potential outcomes. This completeness is key when you determine the likelihood of specific events occurring.
Random Selection
Random selection plays a pivotal role in probability because it ensures fairness in how outcomes are chosen. In the playlist scenario, when you randomly choose a song, each track from the 100 songs has an equal chance of being selected.

This is because, with random selection, all elements of your sample space have the same "weight." Hence, no song is favored over another, making the selection process unbiased.
  • Equal Chance: Every song has a \( \frac{1}{100} \) probability of being played.
  • Fair Play: Random selection prevents predictability. Each outcome remains independent of previous selections.
It's this process that forms the backbone of probability calculations, ensuring that probabilities derived from it are representative and accurate.
Statistical Calculation
The process of calculating probability involves straightforward mathematical steps. It enables you to find the likelihood of drawing a specific song from the playlist. When we talk about statistical calculation in probability, we refer to using fractions or percentages to represent how likely an event is.

In this case, we determined that the probability of choosing a song by your favorite artist is calculated as follows:

Calculate Probability of a Favorite Song

The probability is obtained by dividing the number of favorite songs by the total number of songs:\[\text{Probability} = \frac{15}{100} = 0.15\]

Calculate Probability of a Non-Favorite Song

We find the probability of not selecting a favorite by using the complement rule. By subtracting the probability of selecting a favorite song from 1, you get:\[\text{Probability} = 1 - 0.15 = 0.85\]
These calculations provide a numeric representation of how likely each event is and are the essence of using statistics to understand randomness and chance.

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