Question

Let \(x\) be the number of successes observed in a sample of \(n=5\) items selected from a population of \(N=10 .\) Suppose that of the \(N=10\) items, \(M=6\) are considered "successes." Find the probabilities in Exercises \(11-13 .\) The probability of observing no successes.

Short answer

Answer: The probability of observing no successes in a sample of 5 items is 0.

Step-by-step solution

Identify the known values

In this problem, these are given: - x (number of successes in the sample) = 0 - n (number of items selected) = 5 - N (population size) = 10 - M (number of successes in the population) = 6

Apply the hypergeometric distribution formula

We will plug in the values from Step 1 into the formula: P(0) = \(\frac{\binom{6}{0}\binom{10-6}{5-0}}{\binom{10}{5}}\)

Compute the binomial coefficients

\(\binom{6}{0} = 1\) \(\binom{10-6}{5-0} = \binom{4}{5} = 0\) (since 5 > 4) \(\binom{10}{5} = 252\)

Substitute the binomial coefficients into the formula

P(0) = \(\frac{1 \cdot 0}{252}\)

Calculate the probability

P(0) = \(\frac{0}{252} = 0\) So the probability of observing no successes in a sample of 5 items is 0.

Key concepts

Understanding Probability

Probability is a measure of the likelihood that a particular event will occur. It's a fundamental concept in statistics and it's used in various forms of data analysis and real-world decision making. In the context of the hypergeometric distribution used in the exercise, the probability tells us how likely it is to observe a specific number of successes when we randomly select a sample from a population.

In the given exercise, we're interested in the likelihood of seeing no successes in our sample. We calculate this probability using the hypergeometric distribution formula. The result indicates the chance of a particular outcome happening, which, as we determined in this case, is zero. This means that it's impossible, given our scenario, to not have any successes in our sample based on the population and sample sizes.

Decoding Binomial Coefficients

Binomial coefficients, often represented by \( \binom{n}{k} \), are an integral part of combinatorics, which is the study of counting. They tell us the number of ways we can choose a subset of items from a larger set, regardless of the order of the items. This concept is crucial for calculating probabilities in problems involving combinations, like the hypergeometric distribution problem.

In the solution, we calculated several binomial coefficients to find out how many ways we could select a subset of 'successes' and 'failures' from the population. One of those coefficients, \( \binom{4}{5} \), turned out to be zero, which makes sense because it's impossible to pick 5 items from a set of only 4. Understanding how binomial coefficients work and how to calculate them is key for solving this type of probability problem.

The Role of Statistical Sampling in Probability

Statistical sampling is the process of selecting a subset (a sample) of individuals from within a statistical population to estimate characteristics of the whole population. There are different types of sampling methods, and the hypergeometric distribution arises specifically in scenarios without replacement, meaning once an item is selected, it cannot be chosen again.

The exercise provides a classic example of statistical sampling where we have a finite population and we want to understand the probability of a certain outcome when we cannot sample with replacement. It illustrates the importance of sampling in practical scenarios like quality control or wildlife population estimation. By using sampling, we are able to make informed guesses about larger populations based on smaller, more manageable subsets.