Question

Car color preferences change over the years and according to the particular model that the customer selects. In a recent year, suppose that \(10 \%\) of all luxury cars sold were black. If 25 cars of that year and type are randomly selected, find the following probabilities: a. At least five cars are black. b. At most six cars are black. c. More than four cars are black. d. Exactly four cars are black. e. Between three and five cars (inclusive) are black. f. More than 20 cars are not black

Short answer

b) What is the probability that at most six cars are black? c) What is the probability that more than four cars are black? d) What is the probability that exactly four cars are black? e) What is the probability that between three and five cars (inclusive) are black? f) What is the probability that more than 20 cars are not black?

Step-by-step solution

Identify the Variables

In this problem, we have a binomial distribution situation, where we need to use n and p. Here, n = 25 and p = 0.1.

Calculate Probabilities for Each Case

We will use the binomial probability formula to solve for the probability of each scenario. a. At least five cars are black: We can rewrite this as \(P(k \geq5) = 1 - P(k<5)\) \(P(k<5) = P(0) + P(1) + P(2) + P(3) + P(4)\) Compute each probability using the binomial probability formula and add them up. b. At most six cars are black: We can rewrite this as \(P(k \leq6)\) \(P(k \leq6) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) + P(6)\) Compute each probability using the binomial probability formula and add them up. c. More than four cars are black: This is the same as part (a). At least five cars are black means more than four cars are black. d. Exactly four cars are black: We compute this directly using the binomial probability formula, \(P(k=4)\). e. Between three and five cars (inclusive) are black: We can rewrite this as \(P(3 \leq k \leq 5)\) \(P(3 \leq k \leq 5) = P(3) + P(4) + P(5)\) Compute each probability using the binomial probability formula and add them up. f. More than 20 cars are not black: We can rewrite this as \(P(k \leq 4)\) \(P(k \leq 4) = P(0) + P(1) + P(2) + P(3) + P(4)\) Compute each probability using the binomial probability formula and add them up. This is the same as the result of \(P(k<5)\) from part (a).

Key concepts

Probability Calculation

Probability calculation is a powerful tool used to determine the likelihood of various outcomes when dealing with uncertainties. In this exercise, we focus on the probability of different numbers of black luxury cars among a sample. Probability allows us to quantify how likely it is for a specific event to occur.

There are two key components in calculating these probabilities: the number of trials and the probability of success in a single trial. In our example, the number of trials, denoted as \( n \), is 25, which represents the 25 cars. The probability of success, denoted as \( p \), is 0.1, representing the 10% chance that a luxury car is black. These components are used within the binomial distribution formula:

\[ P(k) = \binom{n}{k} p^k (1-p)^{n-k} \]

Here, \( P(k) \) is the probability of exactly \( k \) successes, \( \binom{n}{k} \) is the binomial coefficient, and \( (1-p) \) is the probability of not succeeding in a trial.

Car Color Preferences

Car color preferences can vary greatly depending on trends, geographic locations, and personal taste. When it comes to luxury cars, black has always been a popular choice due to its association with elegance and sophistication.

In this exercise, it is stated that 10% of luxury cars sold in a particular year were black. Understanding this preference helps analysts and manufacturers predict demand and plan effective marketing strategies. It also gives insights into consumer behavior which can be influential in production decisions. When assessing the preference for black luxury cars versus other colors, companies can stay ahead in the competitive automotive market. Understanding these trends is a crucial part of making informed business strategies.

Luxury Cars

Luxury cars are not only about providing a mode of transportation; they're about the experience, comfort, and status they confer. These vehicles are often equipped with premium features and built with superior craftsmanship, targeting a specific audience who seek quality and exclusivity.

The popularity of luxury cars and their features, including color, can significantly impact purchasing decisions. With black being a classical choice that exudes a certain aura of prestige and sophistication, it reflects the taste and social status of owners. The trends in luxury car preferences, like color, can shift perceptions and sales, influencing how auto companies design and market new models.

For statisticians and marketers, understanding these preferences helps forecast future trends in luxury car sales. Luxury car manufacturers also focus on innovation and design to stay ahead of competitors and meet consumer expectations.

Statistical Analysis

Statistical analysis is a foundational method for interpreting data and making decisions based on collected information. In our scenario, it involves analyzing the probability of different numbers of black luxury cars in a sample.

By applying statistical methods such as the binomial distribution, one can estimate probabilities for various outcomes. For example, determining the likelihood of exactly four black cars or more than four black cars out of a sample of 25 luxury cars. These probabilities aid in drawing conclusions regarding consumer preferences and assessing how likely certain sales scenarios are.

Statistical analysis provides objective insights, reducing reliance on intuition alone. It helps adjust strategies, predict future trends, and ultimately make informed business decisions. In car manufacturing and marketing, leveraging statistical insights is crucial for aligning with consumer needs and expectations.