Question

Suppose a playlist on an MP3 music player consisting of 100 songs includes 8 by a particular artist. Suppose that songs are played by selecting a song at random (with replacement) from the playlist. The random variable \(x\) represents the number of songs until a song by this artist is played. a. Explain why the probability distribution of \(x\) is not binomial. b. Find the following probabilities. (Hint: See Example 6.31) i. \(p(4)\) ii. \(P(x \leq 4)\) iii. \(P(x>4)\) iv. \(P(x \geq 4)\) c. Interpret each of the probabilities in Part (b) and explain the difference between them.

Short answer

The random variable \(x\) represents a geometric distribution as it represents the number of trials until the first success. The calculated probabilities lead to an understanding that there's a 6.7% chance that the first song by the artist is the 4th song played, a 26.9% chance that the first song by the artist is within the first four songs played, a 73.1% chance that the first song by the artist is not within the first four songs played, and the same percentage for the first song by the artist being the 4th or later.

Step-by-step solution

Identify the Type of Distribution

The random variable \(x\) represents the number of songs until a song by this artist is played. This is not a binomial distribution, this is a geometric distribution which describes the number of attempts it would take to get to the first successful outcome.

Calculate Probabilities

For a geometric distribution, the formula for calculating probability is \( p(x) = (1 - p)^{k-1} * p \), where \( p \) is the probability of success, \( k \) is the number of trials and \( x \) is the specific outcome. Here, \( p \) is the probability of playing a song by that artist, which is \(\frac{8}{100}\) or 0.08. i. The probability for \( p(4) \), meaning the first song by the artist is the 4th song played: \((1 - 0.08)^{4-1} * 0.08 = 0.067 \). ii. The probability for \( P(x \leq 4) \), meaning the first song by the artist is one of the first 4 songs played: \( P(x=1) + P(x=2) + P(x=3) + P(x=4) = 0.08(1-(1-0.08)^1) + 0.08(1-(1-0.08)^2) + 0.08(1-(1-0.08)^3) + 0.08(1-(1-0.08)^4) = 0.269 \). iii. The probability for \( P(x > 4) \), meaning the first song by the artist is not within the first 4 songs played: \( 1 - P(x \leq 4) = 1 - 0.269 = 0.731 \). iv. The probability for \( P(x \geq 4) \), meaning the first song by the artist is the 4th song or later: Similar to the previous probability, because \( P(x \geq 4) = P(x > 3) = P(x > 4) + P(x=4) = 0.731 \).

Interpret Probabilities

i. There is a 0.067 or 6.7% chance that the first song by this artist is the 4th song played. ii. There is a 0.269 or 26.9% chance that the first song by this artist is one of the first 4 songs played. iii. There is a 0.731 or 73.1% chance that the first song by this artist is not within the first 4 songs played. iv. There is a 0.731 or 73.1% chance that the first song by this artist is the 4th song played or any song played after the 4th song.

Key concepts

Probability Distribution

When we talk about probability distribution, we refer to a mathematical function that provides the probabilities of occurrence of different possible outcomes for an experiment. It's a system to show all the potential results for a random variable along with their respective probabilities.

Imagine you roll a die; the probability distribution would list the likelihood of rolling a one, a two, and so on, all the way up to a six. In mathematical terms, if the random variable we are considering is a discrete one, meaning it can take on countable values, then the distribution is described by a probability mass function. Conversely, for a continuous random variable, we use a probability density function.

The importance of understanding a probability distribution lies in its power to predict the chance of any outcome occurring, making it a foundational concept in both statistics and real-world decision making. Distributions can be uniform, meaning every outcome has the same likelihood; binomial, for a fixed number of independent yes/no trials each with the same probability of success; or like our example of a playlist, geometric among others. The geometric probability distribution, in particular, is unique in that it represents the number of trials up until the first success.

Random Variable

A random variable is a rule that assigns a number to each outcome of a random circumstance. Essentially, it's a quantitative label that helps summarize the results of random events. For example, if we flip a coin, we might define our random variable to be 0 if we get tails and 1 if we get heads. This helps turn abstract random events into concrete numerical values.

In the case of the MP3 player, the random variable (\(x\)) captures the number of songs played until a track by a specific artist begins. We use random variables because they enable us to quantify and hence calculate the probability of events, especially when dealing with complex scenarios like predicting weather patterns or calculating insurance premiums. The random variable can be discrete or continuous, and as seen in the exercise, it is crucial in shaping the type of probability distribution we rely on.

Geometric Probability

Delving into the realm of geometric probability, we uncover a distribution that is rather specific and unique. It unveils the likelihood that the first event of interest will occur on the \(k\text{-th}\) trial. The geometric distribution is exclusively focused on scenarios where you have a sequence of independent Bernoulli trials - these are experiments with exactly two possible outcomes like success and failure.

In our playlist scenario, the 'success' is selecting a song by a particular artist. What makes the geometric distribution distinct is its 'memoryless' property, which means the probability of success on a given trial is the same regardless of how many trials precede it. As with our example, if we haven't heard a song by the artist after several songs, the probability of the next song being by that artist remains unchanged. This concept is crucial in various fields such as reliability testing of electronics, where manufacturers might be interested in the expected number of uses before the first failure occurs.