Question

Voter Registration A city ward consists of 200 registered voters of whom 125 are registered Republicans and 75 are registered with other parties. On voting day, \(n=10\) people are selected at random for an exit poll in this ward. a. What is the probability distribution, \(p(x),\) for \(x,\) the number of Republicans in the poll? b. Find \(p(10)\). c. Find \(p(0)\).

Short answer

Also, find the probability of having all 10 people in the poll being Republicans and the probability of having no Republicans in the poll.

Step-by-step solution

Identify the relevant hypergeometric distribution parameters

The hypergeometric distribution has three parameters: 1. The population size (N): In this case, the population size is the total number of registered voters in the city ward, which is 200. 2. The number of successes in the population (K): Here, the successes are the number of registered Republicans, which is 125. 3. The sample size (n): For this exit poll, the sample size is 10. We will denote the number of Republicans in the poll by \(x\) and the probability distribution by \(p(x)\).

Write the formula for the hypergeometric distribution

The probability distribution function for the hypergeometric distribution is given by: \(p(x) = \frac{\binom{K}{x} \binom{N-K}{n-x}}{\binom{N}{n}}\) Where \(x\) is the number ofRegistered Republicans in the poll, \(K\) is the total number ofRegistered Republicans, \(N\) is the total number of registered voters, and \(n\) is the size of the sample. We will use this formula to calculate the probability distribution for \(x\).

Find the probability distribution, \(p(x)\), for \(x=0,1,…,10\)

Using the formula from Step 2, we will evaluate the probability distribution for \(x=0\) to \(x=10\): \(p(x) = \frac{\binom{125}{x} \binom{75}{10-x}}{\binom{200}{10}}\) Once we've calculated p(x) for each value of \(x\), we'll have the complete probability distribution for the number ofRepublicans in the poll.

Find \(p(10)\)

To find the probability of having all 10 people in the poll being Republicans, we can plug in \(x=10\) into the formula: \(p(10) = \frac{\binom{125}{10} \binom{75}{0}}{\binom{200}{10}}\) Once we've calculated this probability, we'll have the value of \(p(10)\).

Find \(p(0)\)

To find the probability of having no Republicans in the poll, we can plug in \(x=0\) into the formula: \(p(0) = \frac{\binom{125}{0} \binom{75}{10}}{\binom{200}{10}}\) Once we've calculated this probability, we'll have the value of \(p(0)\).

Key concepts

Probability Distribution

In the realm of statistics, a probability distribution is a mathematical function that provides the probabilities of occurrence of different possible outcomes in an experiment. It is a foundational concept which maps out the likelihood of each possible result, based on a set of parameters.
In the context of our voting day exit poll example, the type of probability distribution we are exploring is called a hypergeometric distribution. This distribution occurs when we perform random sampling without replacement from a finite population, which means that once a voter is chosen for the exit poll, they cannot be selected again. The hypergeometric distribution assigns a probability to every possible number of 'successes' (in our case, the number of registered Republicans) in the sample.
By defining what a 'success' is (selecting a registered Republican), we can use the formula provided to calculate the probability of getting a certain amount of successes in our sample of 10 people. It's crucial to understand that the hypergeometric distribution differs from other distributions, such as the binomial distribution, because it accounts for the decreasing population size after each draw, affecting the probabilities of subsequent draws.

Binomial Coefficient

The binomial coefficient is represented by \(\binom{n}{k}\) which is a key element in combinations and counting problems in mathematics. Intuitively, it tells us the number of ways we can choose k items from a larger set of n distinct items.
In layman's terms, the binomial coefficient calculates the different possibilities of grouping items without considering the sequence in which they are arranged. In the formula for the hypergeometric distribution, we see binomial coefficients being utilized to ascertain the number of ways to select x Republicans from the total of K Republicans (registered ones) and to select the remaining participants of the poll from those not registered as Republicans.
One can calculate the binomial coefficient using factorials, where \(\binom{n}{k} = \frac{n!}{k!(n-k)!}\) and '!' denotes the factorial operation. The factorial of a number n (represented as n!) is the product of all positive integers less than or equal to n. This concept is pivotal because it forms the building blocks of the hypergeometric distribution's formula, enabling us to understand the combination-based nature of the distribution.

Random Sampling

Random sampling is a critical method in statistics where a subset of individuals (a sample) is selected from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process.
In the case of our exit poll, random sampling is conducted when picking 10 people from the 200 registered voters. It is essential in obtaining a representation that reflects the diversity of the whole population without bias. When we say that the sampling is done 'without replacement,' it means once a voter is sampled for the exit poll, they aren't put back into the pool of potential candidates for selection. This is important for the hypergeometric distribution since the odds of selecting a Republican (or others) changes with each person polled, affecting the probability of the subsequent selections.
Random sampling ensures the fairness of statistical analysis, and when done correctly, it allows the researcher to infer the properties of the whole population based on the sample. This concept is central to many studies and experiments that aim to draw conclusions about populations, whether in social sciences, natural sciences, or other fields.