Calculating The Probability Of Selecting A Person With Blue Eyes
In probability theory, determining the likelihood of an event is a fundamental concept. This article explores how to calculate the probability of selecting a person with blue eyes from a group of 1,200 individuals with varying eye colors. We will delve into the basic principles of probability, apply them to the given scenario, and arrive at the correct solution. Understanding these concepts is crucial not only for mathematical problem-solving but also for real-world applications in fields like statistics, data analysis, and decision-making.
Understanding Probability
Probability, at its core, is a measure of the likelihood that an event will occur. It is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. In simpler terms, probability helps us understand how likely something is to happen. The basic formula for calculating probability is:
Probability of an Event = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
This formula is the cornerstone of probability calculations and will be instrumental in solving our problem. To truly grasp this, let's break down each component:
- Number of Favorable Outcomes: This refers to the number of outcomes that align with the event we're interested in. For example, if we're interested in the probability of drawing a heart from a deck of cards, the number of favorable outcomes would be 13 (since there are 13 hearts in a standard deck).
- Total Number of Possible Outcomes: This represents the total number of possible results in the given scenario. In the card example, the total number of possible outcomes would be 52 (the total number of cards in a deck).
Applying Probability to Eye Color
In our scenario, we are looking to determine the probability of selecting a person with blue eyes from a group. Therefore, we need to identify the number of people with blue eyes (favorable outcomes) and the total number of people in the group (total possible outcomes). This foundational understanding will allow us to apply the probability formula effectively.
The application of probability extends far beyond simple calculations; it is a vital tool in various fields. For instance, in statistics, probability is used to make inferences about populations based on sample data. In finance, it is crucial for assessing investment risks and returns. In healthcare, probability helps in understanding the likelihood of disease outbreaks and the effectiveness of treatments. The versatility of probability makes it an indispensable concept in both academic and practical settings.
Problem Setup: Eye Color Distribution
To solve the probability question effectively, let's carefully organize the information provided. We are given a group of 1,200 people, and we know the distribution of eye colors within this group:
- 468 people have brown eyes.
- 252 people have blue eyes.
- 132 people have green eyes.
- 348 people have black eyes.
Our goal is to find the probability that a person chosen at random from this group has blue eyes. The key here is to recognize that the number of people with blue eyes represents our favorable outcomes, while the total number of people represents the total possible outcomes. This understanding is crucial for correctly applying the probability formula.
Identifying Favorable and Total Outcomes
- Favorable Outcomes: The number of people with blue eyes is 252. This is the number of outcomes that satisfy our condition—selecting a person with blue eyes.
- Total Possible Outcomes: The total number of people in the group is 1,200. This represents all possible outcomes when selecting a person at random.
Having identified these two key numbers, we are now ready to apply the probability formula. This step-by-step approach ensures that we understand the underlying concepts and can accurately calculate the probability. This methodical approach is essential for tackling probability problems and ensures clarity and accuracy in our calculations.
Calculating the Probability
Now that we have identified the number of favorable outcomes (people with blue eyes) and the total number of possible outcomes (total people in the group), we can calculate the probability using the formula:
Probability of an Event = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes)
In our case:
- Number of Favorable Outcomes = 252 (people with blue eyes)
- Total Number of Possible Outcomes = 1,200 (total people)
Plugging these values into the formula, we get:
Probability (Blue Eyes) = 252 / 1,200
Simplifying the Fraction
To make the probability easier to understand, we can simplify the fraction. Both 252 and 1,200 are divisible by several numbers. We can start by dividing both by their greatest common divisor, or we can simplify incrementally. Let's simplify incrementally by dividing both numbers by 12:
252 / 12 = 21 1,200 / 12 = 100
So, our simplified fraction becomes:
Probability (Blue Eyes) = 21 / 100
Converting to Percentage
To express the probability as a percentage, we multiply the fraction by 100:
(21 / 100) * 100 = 21%
Therefore, the probability that a person chosen at random from this group has blue eyes is 21%. This final step of converting the fraction to a percentage provides a more intuitive understanding of the probability, making it easier to grasp the likelihood of the event.
Answer and Conclusion
After calculating the probability, we found that the probability of selecting a person with blue eyes from the group of 1,200 people is 21%. This means that if we were to randomly select a person from this group, there is a 21% chance that they would have blue eyes. This result aligns with the principles of probability, where we calculate the likelihood of an event by dividing the number of favorable outcomes by the total number of possible outcomes.
Final Answer
The probability that a person chosen at random from this group has blue eyes is:
P(blue) = 21%
This problem illustrates a fundamental application of probability theory in a real-world scenario. By understanding the basic concepts and applying the probability formula, we can effectively determine the likelihood of various events. This skill is invaluable in a wide range of fields, from statistics and data analysis to everyday decision-making. The ability to calculate and interpret probabilities enables us to make informed choices and predictions based on available data.
Key Takeaways
- Probability is a measure of the likelihood of an event occurring, ranging from 0 (impossible) to 1 (certain).
- The basic probability formula is: Probability of an Event = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes).
- In this problem, the favorable outcome was selecting a person with blue eyes, and the total possible outcomes were the total number of people in the group.
- Simplifying fractions and converting them to percentages can make probabilities easier to understand.
- Probability is a versatile tool with applications in various fields, including statistics, finance, and healthcare.
By mastering these concepts, you can confidently tackle probability problems and apply them to real-world situations. This problem-solving approach not only enhances mathematical skills but also cultivates critical thinking and analytical abilities, which are essential in numerous aspects of life.
