Calculating Probability Of Urban Population In A Sample Of Americans

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In this article, we delve into the realm of probability calculations, specifically focusing on the likelihood of finding a certain number of individuals living in urban areas within a randomly selected group. The claim at hand is that 79% of all Americans reside in cities with populations exceeding 100,000. Based on this premise, we aim to determine the probability of selecting a group of 43 Americans and finding exactly 34 of them living in such urban centers. This exploration involves applying the principles of probability distributions, offering a practical example of how statistical concepts can be used to analyze population data and make predictions.

The core question we address is: If 79% of Americans live in cities with a population greater than 100,000, what is the probability that, in a random sample of 43 Americans, exactly 34 of them reside in these urban areas? This problem falls under the domain of binomial probability, where we have a fixed number of trials (43 Americans), each trial is independent, there are only two outcomes (lives in a city with a population > 100,000 or does not), and the probability of success (living in a city with a population > 100,000) is constant (79%). This intricate calculation helps us understand how likely it is to observe a specific distribution of urban dwellers within a sample, given the overall urban population percentage. The insights gained from such calculations are valuable in various fields, from urban planning to sociological research, where understanding population distributions is crucial.

To tackle this problem, we turn to the binomial probability formula, a cornerstone of statistical analysis for scenarios involving a fixed number of independent trials with binary outcomes. The formula is expressed as:

P(X=k)=(nk)∗pk∗(1−p)(n−k)P(X = k) = \binom{n}{k} * p^k * (1 - p)^{(n - k)}

Where:

  • P(X=k)P(X = k) is the probability of observing exactly kk successes.
  • (nk)\binom{n}{k} is the binomial coefficient, representing the number of ways to choose kk successes from nn trials, calculated as n!/(k!(n−k)!)n! / (k!(n - k)!).
  • nn is the total number of trials.
  • kk is the number of successes we are interested in.
  • pp is the probability of success on a single trial.
  • (1−p)(1 - p) is the probability of failure on a single trial.

In our specific scenario:

  • n=43n = 43 (the number of Americans randomly selected)
  • k=34k = 34 (the number of Americans we want to live in cities with a population > 100,000)
  • p=0.79p = 0.79 (the probability that an American lives in a city with a population > 100,000)

Applying this formula allows us to precisely calculate the probability of finding exactly 34 urban residents in our sample of 43, providing a quantitative answer to our problem statement. The binomial coefficient, also known as “n choose k,” plays a critical role in accounting for all possible combinations of successes and failures, ensuring an accurate probability assessment.

Let's break down the calculation process step by step:

  1. Calculate the binomial coefficient: (4334)=43!34!(43−34)!=43!34!9!\binom{43}{34} = \frac{43!}{34!(43 - 34)!} = \frac{43!}{34!9!} This step determines the number of ways to choose 34 people out of 43. Calculating factorials of large numbers can be computationally intensive, but tools and calculators are readily available to assist.
  2. Calculate the probability of 34 successes: pk=0.7934p^k = 0.79^{34} This step calculates the probability of 34 Americans living in urban areas, assuming the probability of any one American living in such an area is 0.79.
  3. Calculate the probability of 9 failures: (1−p)(n−k)=(1−0.79)(43−34)=0.219(1 - p)^{(n - k)} = (1 - 0.79)^{(43 - 34)} = 0.21^9 This step calculates the probability of the remaining 9 Americans not living in urban areas. It's essential to consider the probability of failure to complete the binomial probability calculation.
  4. Combine the results: P(X=34)=(4334)∗0.7934∗0.219P(X = 34) = \binom{43}{34} * 0.79^{34} * 0.21^9 This final step multiplies the binomial coefficient by the probabilities of success and failure to find the overall probability.

Each step is crucial in arriving at the final probability, and understanding the logic behind each step is vital for applying the binomial probability formula correctly. Accurate calculation is key to deriving meaningful insights from the data.

Now, let's proceed with the detailed calculations:

  1. Binomial Coefficient Calculation:

    (4334)=43!34!9!=43×42×41×40×39×38×37×36×359×8×7×6×5×4×3×2×1=486,204,120\binom{43}{34} = \frac{43!}{34!9!} = \frac{43 \times 42 \times 41 \times 40 \times 39 \times 38 \times 37 \times 36 \times 35}{9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1} = 486,204,120

    The binomial coefficient represents the number of ways to select 34 individuals from a group of 43, a significantly large number indicating the numerous possible combinations.

  2. Probability of 34 Successes:

    0.7934≈0.0002250.79^{34} \approx 0.000225

    This is the probability that 34 randomly selected individuals all live in cities with populations greater than 100,000. The probability is relatively small because it requires a specific sequence of events.

  3. Probability of 9 Failures:

    0.219≈1.95×10−70.21^9 \approx 1.95 \times 10^{-7}

    This represents the probability that the remaining 9 individuals do not live in such cities. This probability is extremely small, reflecting the rarity of this particular outcome.

  4. Final Probability Calculation:

    P(X=34)=486,204,120×0.000225×1.95×10−7≈0.0213P(X = 34) = 486,204,120 \times 0.000225 \times 1.95 \times 10^{-7} \approx 0.0213

    The final probability, approximately 0.0213, represents the likelihood of finding exactly 34 individuals living in urban areas in a sample of 43 Americans, given the overall urban population percentage of 79%. This value provides a concrete answer to our initial question.

After performing the calculations, we find that the probability of exactly 34 out of 43 randomly selected Americans living in cities with a population greater than 100,000 is approximately 0.0213, or 2.13%. This result is crucial for several reasons. First, it quantitatively answers our initial question, providing a specific probability value. Second, it allows us to interpret the likelihood of observing such an event. A probability of 2.13% suggests that this is a relatively unlikely occurrence. In practical terms, if we were to repeatedly sample 43 Americans, we would expect to find exactly 34 of them living in large cities in only about 2.13% of the samples.

This interpretation is vital for understanding the implications of the initial claim that 79% of Americans live in urban areas. If we were to conduct an actual survey and found significantly fewer than 34 urban residents in our sample, it might suggest that the initial claim is inaccurate or that our sample is not truly random. Conversely, if we found a number close to 34, it would support the claim. The probability we calculated serves as a benchmark against which to compare empirical observations. This principle is fundamental in statistical hypothesis testing, where we use sample data to make inferences about population parameters.

The result of this probability calculation has several implications and applications across various fields. In urban planning, understanding the distribution of urban and rural populations is crucial for resource allocation, infrastructure development, and policy-making. Knowing the probability of finding a certain number of urban residents in a sample can help planners assess the adequacy of services and infrastructure in different regions. For example, if a planner needs to determine the optimal location for a new hospital, they might consider the urban population density and the likelihood of patients coming from urban areas.

In sociological research, this type of calculation can be used to study demographic trends and patterns. Researchers might be interested in understanding how urban populations are changing over time and what factors contribute to these changes. By comparing probabilities calculated from different datasets, sociologists can gain insights into migration patterns, urbanization rates, and other social phenomena. Moreover, in market research, businesses can use these probabilities to target their products and services more effectively. For instance, a company selling urban-specific products might use population data to identify areas with a high concentration of urban residents and tailor their marketing efforts accordingly.

Public health is another area where these calculations are valuable. Understanding the distribution of populations in urban versus rural settings is critical for assessing health disparities and planning public health interventions. For example, if a particular health issue is more prevalent in urban areas, public health officials can use population data to target interventions to the most at-risk communities. The ability to accurately calculate and interpret probabilities related to population distributions is a fundamental skill in many professional fields, contributing to informed decision-making and effective policy implementation.

In conclusion, we have successfully calculated the probability of finding exactly 34 out of 43 randomly selected Americans living in cities with populations greater than 100,000, given the claim that 79% of all Americans reside in such urban areas. By applying the binomial probability formula, we determined that the probability is approximately 0.0213, or 2.13%. This result highlights the importance of understanding probability distributions and their applications in analyzing population data.

The ability to calculate such probabilities is not just an academic exercise; it has practical implications across a wide range of fields. From urban planning and sociological research to market research and public health, understanding population distributions is essential for making informed decisions and developing effective strategies. The binomial probability formula, in particular, provides a powerful tool for analyzing scenarios with binary outcomes, allowing us to quantify the likelihood of specific events and make data-driven predictions.

Moreover, this exercise underscores the significance of statistical reasoning in everyday life. By understanding basic probability concepts, we can better interpret statistical claims and make sense of the world around us. Whether we are evaluating the results of a survey, assessing the effectiveness of a public health intervention, or making business decisions, a solid understanding of probability is crucial for informed decision-making. The principles we have explored in this article serve as a foundation for further statistical analysis and critical thinking.