Hypergeometric Probability Calculation Calculate P(X=7) With N=120, N=23, And K=25

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In the realm of probability and statistics, the hypergeometric distribution stands out as a crucial tool for analyzing scenarios involving sampling without replacement. This distribution is particularly relevant when we need to determine the probability of obtaining a specific number of successes in a sample drawn from a finite population. Unlike the binomial distribution, which assumes independent trials, the hypergeometric distribution accounts for the changing probabilities that arise when items are drawn without being returned to the population. This article delves into the intricacies of a hypergeometric experiment, providing a comprehensive guide on how to calculate probabilities using the hypergeometric formula. We will focus on a specific example where the population size (N) is 120, the sample size (n) is 23, the number of successes in the population (k) is 25, and we are interested in finding the probability of observing exactly 7 successes in our sample (X=7). This detailed exploration will equip you with the knowledge to confidently tackle similar problems and understand the underlying principles of hypergeometric probability.

Understanding hypergeometric experiments is fundamental in various fields, including quality control, genetics, and polling. For instance, in quality control, one might want to determine the probability of selecting a certain number of defective items from a batch without replacing them. In genetics, it could be used to calculate the likelihood of inheriting a specific combination of genes. In polling, the hypergeometric distribution can help estimate the probability of selecting a particular demographic group from a population. The key characteristic of a hypergeometric experiment is that the sampling is done without replacement, meaning once an item is selected, it is not returned to the population, thus affecting the probabilities of subsequent selections. This is what distinguishes it from other distributions like the binomial distribution, where each trial is independent. The hypergeometric distribution allows for a more accurate representation of probabilities in situations where the population size is finite and the sampling is done without replacement, making it an indispensable tool in statistical analysis.

The practical applications of hypergeometric probability extend beyond theoretical exercises, providing real-world insights across diverse industries. Consider a scenario in environmental science where researchers are studying a population of fish in a lake, and they want to estimate the number of fish that are infected with a particular parasite. They capture a sample of fish, examine them for the parasite, and then release them back into the lake. The hypergeometric distribution can be used to calculate the probability of observing a specific number of infected fish in their sample, given the total number of fish in the lake and the estimated number of infected fish. This information can then be used to make informed decisions about the health of the fish population and the potential impact of the parasite. Similarly, in the pharmaceutical industry, hypergeometric probability is used to assess the effectiveness of a new drug. Clinical trials involve selecting a sample of patients from a larger population and administering the drug to them. The hypergeometric distribution can help calculate the probability of observing a certain number of positive responses to the drug in the sample, which is crucial for determining whether the drug is likely to be effective in the broader population. By understanding and applying the principles of hypergeometric probability, professionals in various fields can make data-driven decisions and gain valuable insights into the systems they are studying.

The hypergeometric formula is the cornerstone of calculating probabilities in hypergeometric experiments. It provides a precise way to determine the likelihood of observing a specific number of successes in a sample drawn without replacement from a finite population. The formula is expressed as:

P(X=x)=(kx)(N−kn−x)(Nn)P(X = x) = \frac{{\binom{k}{x} \binom{N - k}{n - x}}}{{\binom{N}{n}}}

Where:

  • P(X = x) is the probability of observing exactly x successes.
  • N is the total population size.
  • n is the sample size.
  • k is the number of successes in the population.
  • x is the number of successes in the sample.
  • (nk)\binom{n}{k} represents the binomial coefficient, calculated as n!k!(n−k)!\frac{n!}{k!(n-k)!}, where ! denotes the factorial function.

Each component of the hypergeometric formula plays a crucial role in accurately determining the probability. The term (kx)\binom{k}{x} calculates the number of ways to choose x successes from the k successes available in the population. This is the numerator's first component, representing the favorable outcomes in terms of successes. The second term in the numerator, (N−kn−x)\binom{N - k}{n - x}, calculates the number of ways to choose the remaining (n - x) items from the (N - k) failures in the population. This ensures we account for the ways to select the non-successes in our sample. Together, these two terms in the numerator give us the total number of ways to achieve the specific outcome of x successes in a sample of size n. The denominator, (Nn)\binom{N}{n}, calculates the total number of ways to choose a sample of size n from the entire population of size N. This represents the total possible outcomes, regardless of the number of successes. By dividing the number of favorable outcomes (numerator) by the total number of possible outcomes (denominator), we arrive at the probability P(X = x) of observing exactly x successes in the sample. Understanding the significance of each component is essential for applying the formula correctly and interpreting the results effectively. The formula provides a powerful tool for analyzing situations where sampling without replacement is involved, offering a precise measure of the likelihood of specific outcomes.

The binomial coefficient, denoted as (nk)\binom{n}{k}, is a critical element within the hypergeometric formula, representing the number of ways to choose k items from a set of n items without regard to order. Its mathematical definition is n!k!(n−k)!\frac{n!}{k!(n-k)!}, where the factorial function, denoted by !, calculates the product of all positive integers up to a given number. For instance, 5! = 5 × 4 × 3 × 2 × 1 = 120. The binomial coefficient is also known as a combination, and it is widely used in combinatorics and probability theory. Understanding how to calculate and interpret binomial coefficients is essential for applying the hypergeometric formula correctly. Let's consider a simple example: suppose we want to choose 3 items from a set of 5 items. The binomial coefficient would be (53)=5!3!(5−3)!=5!3!2!=120(6)(2)=10\binom{5}{3} = \frac{5!}{3!(5-3)!} = \frac{5!}{3!2!} = \frac{120}{(6)(2)} = 10. This means there are 10 different ways to choose 3 items from a set of 5 items. In the context of the hypergeometric formula, the binomial coefficients help us quantify the number of ways to select successes and failures in our sample. They allow us to account for the different combinations of items that can be chosen, ensuring that we accurately calculate the probability of observing a specific outcome. The binomial coefficient's role in the formula highlights the importance of considering combinations rather than permutations when dealing with sampling without replacement, as the order in which items are selected does not affect the outcome.

Now, let's apply the hypergeometric formula to the given parameters: N = 120, n = 23, k = 25, and X = 7. We want to calculate P(X = 7), which is the probability of observing exactly 7 successes in our sample.

  1. Identify the parameters:
    • N = 120 (total population size)
    • n = 23 (sample size)
    • k = 25 (number of successes in the population)
    • x = 7 (number of successes in the sample)
  2. Plug the values into the formula:

    P(X=7)=(257)(120−2523−7)(12023)P(X = 7) = \frac{{\binom{25}{7} \binom{120 - 25}{23 - 7}}}{{\binom{120}{23}}}

    P(X=7)=(257)(9516)(12023)P(X = 7) = \frac{{\binom{25}{7} \binom{95}{16}}}{{\binom{120}{23}}}

  3. Calculate the binomial coefficients:

    • (257)=25!7!(25−7)!=25!7!18!=480,700\binom{25}{7} = \frac{25!}{7!(25 - 7)!} = \frac{25!}{7!18!} = 480,700

    • (9516)=95!16!(95−16)!=95!16!79!=11,696,895,455\binom{95}{16} = \frac{95!}{16!(95 - 16)!} = \frac{95!}{16!79!} = 11,696,895,455

    • (12023)=120!23!(120−23)!=120!23!97!=1.609,477,828,200\binom{120}{23} = \frac{120!}{23!(120 - 23)!} = \frac{120!}{23!97!} = 1.609,477,828,200

  4. Substitute the binomial coefficients back into the formula:

    P(X=7)=480,700imes11,696,895,4551,609,477,828,200P(X = 7) = \frac{{480,700 imes 11,696,895,455}}{{1,609,477,828,200}}

  5. Calculate the probability:

    P(X=7)=5,622,505,905,8351,609,477,828,200≈0.3493P(X = 7) = \frac{{5,622,505,905,835}}{{1,609,477,828,200}} \approx 0.3493

The step-by-step calculation highlights the practical application of the hypergeometric formula and demonstrates the importance of precision in each step. The first step involves identifying the parameters, ensuring that each value is correctly assigned to its corresponding variable. This is crucial because an error in identifying the parameters can lead to an incorrect result. The second step involves plugging the values into the formula, which sets up the equation for the probability calculation. The third step is perhaps the most computationally intensive, as it requires calculating the binomial coefficients. These coefficients represent the number of ways to choose a subset of items from a larger set, and their calculation involves factorials, which can quickly become very large numbers. Using a calculator or software that can handle large numbers is often necessary to perform these calculations accurately. The fourth step involves substituting the calculated binomial coefficients back into the formula, setting up the final calculation. The fifth and final step is to calculate the probability, which involves dividing the product of the numerator binomial coefficients by the denominator binomial coefficient. The result is a probability value between 0 and 1, representing the likelihood of observing the specified number of successes in the sample. By following these steps carefully, one can accurately calculate hypergeometric probabilities and gain valuable insights into the underlying processes being studied. The process illustrates the power of the hypergeometric formula in quantifying uncertainty and making informed decisions based on statistical analysis.

Approximating the final probability to four decimal places, we get P(X = 7) ≈ 0.3493. This means that there is approximately a 34.93% chance of observing exactly 7 successes in a sample of 23, drawn from a population of 120, where there are 25 successes in the population. This probability provides valuable information about the likelihood of this specific outcome and can be used for further analysis and decision-making. Understanding the magnitude of the probability is crucial for interpreting the results of the experiment. A probability close to 1 indicates a very likely outcome, while a probability close to 0 suggests a very unlikely outcome. A probability around 0.5 indicates a roughly even chance of the outcome occurring. In this case, the probability of 0.3493 suggests that observing 7 successes is a moderately likely outcome, but not overwhelmingly so. The result can be compared to other probabilities calculated for different numbers of successes to gain a more complete understanding of the distribution. For example, one might calculate the probability of observing 6 successes or 8 successes to see how the likelihood changes as the number of successes varies. This broader perspective can provide a more nuanced understanding of the underlying process and the factors that influence the observed outcomes. The ability to calculate and interpret hypergeometric probabilities is a valuable skill in various fields, from quality control and genetics to polling and environmental science, enabling professionals to make informed decisions based on statistical evidence.

In conclusion, the hypergeometric distribution is a powerful statistical tool for calculating probabilities in scenarios involving sampling without replacement. By understanding the formula and its components, we can accurately determine the likelihood of observing a specific number of successes in a sample. In the given example, we calculated the probability of observing exactly 7 successes in a sample of 23, drawn from a population of 120, with 25 successes in the population. The calculated probability, P(X = 7) ≈ 0.3493, provides valuable insights into the likelihood of this particular outcome. This detailed exploration of the hypergeometric experiment equips us with the knowledge to tackle similar problems and confidently apply this distribution in various real-world scenarios.