Probability Of Not Selecting A Yellow Car A Comprehensive Guide

In this comprehensive article, we will delve into the concept of probability, specifically focusing on calculating the likelihood of not selecting a yellow car from a car dealership's inventory. This is a fundamental problem in probability theory, often encountered in various real-life scenarios. Understanding how to solve such problems can provide valuable insights into decision-making processes and risk assessment. Let's embark on this mathematical journey to unravel the solution step by step.

A car dealership has a full car lot with a diverse range of colors. Among the vehicles, 19 are red, 22 are blue, 7 are white, and 2 are yellow. Our primary objective is to determine the probability that a randomly selected car from this lot is not yellow. This involves understanding the basic principles of probability and applying them to a specific scenario.

Probability, at its core, is the measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 signifies impossibility and 1 signifies certainty. The probability of an event can be calculated using the formula:

$$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of possible outcomes}}$$

Where:

• $P(E)$ represents the probability of event E occurring.
• "Number of favorable outcomes" refers to the count of outcomes that align with the event in question.
• "Total number of possible outcomes" is the overall count of all potential outcomes.

This foundational understanding of probability is crucial for tackling our car selection problem.

To begin, we need to determine the total number of cars in the dealership's lot. This is done by summing up the counts of cars of each color:

$$\text{Total Cars} = \text{Red Cars} + \text{Blue Cars} + \text{White Cars} + \text{Yellow Cars}$$

Substituting the given values:

$$\text{Total Cars} = 19 + 22 + 7 + 2 = 50$$

Thus, there are a total of 50 cars in the lot. This figure serves as the denominator in our probability calculation.

Next, we need to calculate the number of cars that are not yellow. This can be done by subtracting the number of yellow cars from the total number of cars:

$$\text{Cars Not Yellow} = \text{Total Cars} - \text{Yellow Cars}$$

Substituting the known values:

$$\text{Cars Not Yellow} = 50 - 2 = 48$$

So, there are 48 cars that are not yellow. This value will be the numerator in our probability fraction.

Now, we can calculate the probability of selecting a car that is not yellow. Using the probability formula:

$$P(\text{Not Yellow}) = \frac{\text{Number of Cars Not Yellow}}{\text{Total Number of Cars}}$$

Substituting the values we calculated:

$$P(\text{Not Yellow}) = \frac{48}{50}$$

This fraction represents the probability of not selecting a yellow car. To express this probability as a decimal, we perform the division:

$$P(\text{Not Yellow}) = 0.96$$

Thus, the probability of selecting a car that is not yellow is 0.96.

The problem statement requires us to round our answer to the nearest hundredth. Since our calculated probability is already expressed to the hundredth place (0.96), no further rounding is necessary. The probability remains 0.96.

In conclusion, the probability that a randomly selected car from the dealership's lot is not yellow is 0.96. This result was obtained by systematically applying the principles of probability, including calculating the total number of outcomes and the number of favorable outcomes. This example underscores the practical application of probability in everyday scenarios.

$$P(\text{Not Yellow}) = 0.96$$

Another way to approach this problem is by using the complement rule in probability. The complement rule states that the probability of an event not occurring is equal to 1 minus the probability of the event occurring. In mathematical terms:

$$P(\text{Not } A) = 1 - P(A)$$

Where:

• $P(\text{Not } A)$ is the probability of event A not occurring.
• $P(A)$ is the probability of event A occurring.

In our case, event A is selecting a yellow car. So, we can first calculate the probability of selecting a yellow car and then use the complement rule to find the probability of not selecting a yellow car.

To find the probability of selecting a yellow car, we use the same basic probability formula:

$$P(\text{Yellow}) = \frac{\text{Number of Yellow Cars}}{\text{Total Number of Cars}}$$

We already know that there are 2 yellow cars and a total of 50 cars. Substituting these values into the formula:

$$P(\text{Yellow}) = \frac{2}{50}$$

Simplifying the fraction, we get:

$$P(\text{Yellow}) = \frac{1}{25}$$

Converting this fraction to a decimal:

$$P(\text{Yellow}) = 0.04$$

So, the probability of selecting a yellow car is 0.04.

Now that we have the probability of selecting a yellow car, we can use the complement rule to find the probability of not selecting a yellow car:

$$P(\text{Not Yellow}) = 1 - P(\text{Yellow})$$

Substituting the value we calculated for $P(\text{Yellow})$:

$$P(\text{Not Yellow}) = 1 - 0.04$$

$$P(\text{Not Yellow}) = 0.96$$

As we can see, using the complement rule gives us the same result as our previous method: the probability of not selecting a yellow car is 0.96.

The complement rule is particularly useful when calculating the probability of an event not occurring is easier than calculating the probability of it occurring directly. In some cases, determining the number of outcomes that do not satisfy a condition can be more complex than determining the number of outcomes that do satisfy the condition. The complement rule provides a simpler and more efficient way to find the desired probability.

Understanding probability is not just an academic exercise; it has numerous real-world applications across various fields. Here are a few examples:

1. Insurance: Insurance companies use probability to assess the risk of insuring individuals or assets. They calculate the likelihood of certain events occurring (e.g., accidents, illnesses, natural disasters) and set premiums accordingly.
2. Finance: In finance, probability is used to model investment returns, assess risk, and make informed decisions about buying and selling assets. The stock market, for instance, is heavily influenced by probabilistic models.
3. Healthcare: Probability plays a crucial role in medical research and clinical practice. It is used to determine the effectiveness of treatments, diagnose diseases, and predict patient outcomes.
4. Gambling: The gambling industry is built on probability. Casinos and lotteries use probability to set odds and calculate payouts. Understanding probability can help individuals make informed decisions about gambling.
5. Weather Forecasting: Meteorologists use probability to predict the weather. They analyze historical data and current conditions to estimate the likelihood of rain, snow, or other weather events.
6. Quality Control: Manufacturers use probability to ensure the quality of their products. They sample products from a production line and use statistical methods to assess the likelihood of defects.

These are just a few examples of how probability is used in the real world. Its applications are vast and continue to grow as our understanding of statistics and data analysis advances.

In this article, we have explored the concept of probability and applied it to a practical problem: calculating the probability of not selecting a yellow car from a dealership's lot. We used two different methods to arrive at the same answer, highlighting the versatility of probability theory. We also discussed the complement rule and its benefits, as well as real-world applications of probability. Understanding probability is an essential skill for making informed decisions in a wide range of situations. Whether you are assessing risk, making investments, or simply trying to understand the world around you, probability provides a powerful framework for analysis and decision-making.

This example demonstrates the importance of breaking down a problem into smaller, manageable steps and applying the appropriate formulas and concepts. By understanding the fundamentals of probability, we can tackle more complex problems and gain valuable insights into the likelihood of various outcomes.