Calculating Probability Exactly 2 Voters Support Ballot Initiative

Jul 11, 2025 by ADMIN 67 views

Probability of Exactly Two Voters Supporting a Ballot Initiative

Introduction

In the realm of probability, understanding how to calculate the likelihood of specific outcomes in a series of independent trials is crucial. This article delves into a scenario where we explore the probability of a certain number of voters supporting a ballot initiative, given a specific support rate and a random survey. We will use the binomial probability formula to solve this problem, providing a detailed explanation and calculation.

Problem Statement

According to a recent poll, 30% of voters support a particular ballot initiative. Hans conducts a random survey of 5 voters. Our goal is to determine the probability that exactly 2 of these 5 voters will be in favor of the ballot initiative. We will round the final answer to the nearest thousandth.

Understanding Binomial Probability

The problem described is a classic example of a binomial probability scenario. Binomial probability deals with the likelihood of achieving a specific number of successes in a fixed number of independent trials, where each trial has only two possible outcomes: success or failure. In this case, a “success” is a voter supporting the ballot initiative, and a “failure” is a voter not supporting it.

Key Characteristics of Binomial Probability

Fixed Number of Trials: There is a predetermined number of trials (in our case, 5 voters are surveyed).
Independent Trials: Each trial is independent of the others, meaning the outcome of one voter’s opinion does not affect the others.
Two Possible Outcomes: Each trial results in either success (voter supports the initiative) or failure (voter does not support the initiative).
Constant Probability of Success: The probability of success (p) remains the same for each trial (30% or 0.3 in our case).

The Binomial Probability Formula

The binomial probability formula is used to calculate the probability of obtaining exactly k successes in n independent trials. The formula is as follows:

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

Where:

$P(X = k)$ is the probability of exactly k successes.
$\binom{n}{k}$ is the binomial coefficient, which represents the number of ways to choose k successes from n trials. It is calculated as $\frac{n!}{k!(n - k)!}$ , where ! denotes the factorial function.
$p$ is the probability of success on a single trial.
$n$ is the total number of trials.
$k$ is the number of successes we want to find the probability for.
$(1 - p)$ is the probability of failure on a single trial.
$(n - k)$ is the number of failures.

Applying the Formula to Our Problem

In our scenario:

$n = 5$ (number of voters surveyed)
$k = 2$ (number of voters in favor of the initiative)
$p = 0.3$ (probability of a voter supporting the initiative)
$(1 - p) = 0.7$ (probability of a voter not supporting the initiative)

Let's plug these values into the binomial probability formula:

P(X = 2) = \binom{5}{2} * (0.3)^2 * (0.7)^{(5 - 2)}

Step-by-Step Calculation

Calculate the Binomial Coefficient:
$\binom{5}{2} = \frac{5!}{2!(5 - 2)!} = \frac{5!}{2!3!} = \frac{5 * 4 * 3 * 2 * 1}{(2 * 1)(3 * 2 * 1)} = \frac{120}{(2)(6)} = 10$
So, there are 10 ways to choose 2 voters out of 5.
Calculate the Probability of 2 Successes:
$(0.3)^2 = 0.09$
Calculate the Probability of 3 Failures:
$(0.7)^{(5 - 2)} = (0.7)^3 = 0.343$
Plug the Values into the Formula:
$P(X = 2) = 10 * 0.09 * 0.343$
Calculate the Final Probability:
$P(X = 2) = 10 * 0.09 * 0.343 = 0.3087$

Rounding the Answer

The problem asks us to round the answer to the nearest thousandth. Therefore, we round 0.3087 to 0.309.

Result

The probability that exactly 2 out of the 5 surveyed voters will be in favor of the ballot initiative is approximately 0.309 or 30.9%.

Conclusion

In this article, we successfully calculated the probability of a specific outcome in a binomial scenario. By understanding and applying the binomial probability formula, we determined that there is a 30.9% chance that exactly 2 out of 5 randomly surveyed voters will support the ballot initiative, given a 30% overall support rate. This method is widely applicable in various fields, including statistics, surveys, and data analysis, to assess the likelihood of specific events occurring within a set of independent trials. The binomial probability formula is a powerful tool for understanding and predicting outcomes in situations where there are binary results, making it an essential concept in probability theory.

Further Applications of Binomial Probability

The concepts explored in this article extend beyond just political surveys. Binomial probability is a fundamental tool in many fields, including:

Quality Control: Manufacturers use binomial probability to assess the likelihood of defective products in a batch. For instance, if a production line has a 2% defect rate, the binomial probability can help determine the chance of finding a certain number of defective items in a sample.
Medical Research: In clinical trials, binomial probability is used to evaluate the effectiveness of treatments. Researchers might want to know the probability of a certain number of patients showing improvement after receiving a new medication.
Genetics: Geneticists use binomial probability to predict the likelihood of offspring inheriting specific traits. For example, if both parents carry a recessive gene, binomial probability can help determine the chance of their child expressing that trait.
Marketing: Marketers use binomial probability to estimate the success rate of advertising campaigns. They might analyze the probability of a certain number of customers responding to an ad based on historical data.
Gambling and Games of Chance: Binomial probability is crucial in understanding games of chance, such as coin flips or dice rolls. It can help players assess their odds of winning in various scenarios.

Practical Tips for Using Binomial Probability

When applying the binomial probability formula, keep these tips in mind:

Ensure Independence: Verify that each trial is independent of the others. If the outcome of one trial affects subsequent trials, the binomial probability formula may not be appropriate.
Check for Constant Probability: Ensure that the probability of success remains constant across all trials. If the probability changes, you may need to use a different statistical method.
Use Technology: For large values of n and k, calculating factorials and binomial coefficients can be cumbersome. Utilize calculators or statistical software to simplify the calculations.
Consider Approximations: In some cases, when n is large and p is close to 0.5, the binomial distribution can be approximated by the normal distribution, which can simplify calculations.

Advanced Concepts in Binomial Probability

For those interested in delving deeper into binomial probability, here are a few advanced concepts to explore:

Binomial Distribution: The binomial distribution is the probability distribution of the number of successes in a sequence of independent experiments, each with a binary outcome. It provides a comprehensive view of all possible outcomes and their probabilities.
Mean and Variance: The mean (average) and variance (spread) of a binomial distribution can be calculated using simple formulas. The mean is given by $μ = np$ , and the variance is given by $σ^2 = np(1 - p)$ .
Cumulative Probability: Cumulative probability refers to the probability of observing k or fewer successes. It is calculated by summing the probabilities for all values from 0 to k.
Hypothesis Testing: Binomial probability is used in hypothesis testing to determine whether observed results are statistically significant. For instance, it can be used to test whether the support for a ballot initiative has changed significantly over time.

In summary, binomial probability is a versatile and powerful tool for analyzing situations with binary outcomes. Its applications span a wide range of fields, making it an essential concept for anyone working with data and statistics. By mastering the principles and techniques of binomial probability, you can gain valuable insights into the likelihood of events and make more informed decisions.