According To A Poll, 30 % 30\% 30% Of Voters Support A Ballot Initiative. Hans Randomly Surveys 5 Voters. What Is The Probability That Exactly 2 Voters Will Be In Favor Of The Ballot Initiative? Round The Answer To The Nearest

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Introduction

In the realm of probability and statistics, understanding the likelihood of certain events occurring is crucial. A recent poll revealed that $30\%$ of voters support a ballot initiative. To gain a deeper insight into the public's opinion, Hans decided to conduct a random survey of 5 voters. The question at hand is: what is the probability that exactly 2 voters will be in favor of the ballot initiative? In this article, we will delve into the world of probability and calculate the desired probability.

Understanding the Problem

To begin, let's break down the problem. We have a total of 5 voters, and we want to find the probability that exactly 2 of them will be in favor of the ballot initiative. This means that 3 voters will be against the initiative. The probability of a single voter being in favor of the initiative is $30\%$, or $0.3$. Conversely, the probability of a single voter being against the initiative is $70\%$, or $0.7$.

Using the Binomial Distribution

The binomial distribution is a probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success. In this case, we have 5 trials (voters), and each trial has a probability of success (being in favor of the initiative) of $0.3$. We want to find the probability of exactly 2 successes (voters in favor of the initiative).

The binomial distribution is given by the formula:

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

where:

• $P(X=k)$ is the probability of exactly $k$ successes
• $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time
• $p$ is the probability of success
• $n$ is the number of trials
• $k$ is the number of successes

In our case, $n=5$, $k=2$, and $p=0.3$. Plugging these values into the formula, we get:

$$P(X=2) = \binom{5}{2} (0.3)^2 (1-0.3)^{5-2}$$

Calculating the Probability

Now, let's calculate the probability using the formula above.

$$P(X=2) = \binom{5}{2} (0.3)^2 (1-0.3)^{5-2}$$

$$P(X=2) = 10 \times 0.09 \times 0.7^3$$

$$P(X=2) = 10 \times 0.09 \times 0.343$$

$$P(X=2) = 0.3087$$

Rounding the Answer

The probability of exactly 2 voters being in favor of the ballot initiative is approximately $0.3087$. Rounding this value to the nearest hundredth, we get:

$$P(X=2) \approx 0.31$$

Conclusion

In conclusion, the probability of exactly 2 voters being in favor of the ballot initiative, given that $30\%$ of voters support the initiative, is approximately $0.31$. This value provides valuable insight into the public's opinion on the ballot initiative and can be used to inform decision-making.

Future Research Directions

Future research directions could include:

• Investigating the impact of different sample sizes on the probability of exactly 2 voters being in favor of the initiative
• Examining the relationship between the probability of exactly 2 voters being in favor of the initiative and other demographic factors, such as age or income level
• Developing more advanced statistical models to account for potential biases and confounding variables in the data

Limitations of the Study

This study has several limitations. Firstly, the sample size is relatively small, which may limit the generalizability of the results. Secondly, the study assumes that the probability of a single voter being in favor of the initiative is $30\%$, which may not be accurate in reality. Finally, the study does not account for potential biases and confounding variables in the data.

Recommendations for Future Research

Based on the findings of this study, we recommend the following:

• Conducting a larger-scale study to increase the sample size and improve the generalizability of the results
• Collecting more detailed demographic data to examine the relationship between the probability of exactly 2 voters being in favor of the initiative and other demographic factors
• Developing more advanced statistical models to account for potential biases and confounding variables in the data

Conclusion

In conclusion, this study provides valuable insights into the probability of exactly 2 voters being in favor of the ballot initiative, given that $30\%$ of voters support the initiative. The results of this study can be used to inform decision-making and provide a foundation for future research in this area.

Introduction

In our previous article, we explored the probability of exactly 2 voters being in favor of a ballot initiative, given that $30\%$ of voters support the initiative. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the probability of exactly 2 voters being in favor of the ballot initiative?

A: The probability of exactly 2 voters being in favor of the ballot initiative is approximately $0.31$.

Q: How was the probability calculated?

A: The probability was calculated using the binomial distribution formula, which takes into account the number of trials (voters), the probability of success (being in favor of the initiative), and the number of successes (voters in favor of the initiative).

Q: What is the binomial distribution formula?

A: The binomial distribution formula is:

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

where:

• $P(X=k)$ is the probability of exactly $k$ successes
• $\binom{n}{k}$ is the number of combinations of $n$ items taken $k$ at a time
• $p$ is the probability of success
• $n$ is the number of trials
• $k$ is the number of successes

Q: What is the significance of the binomial distribution in this context?

A: The binomial distribution is significant in this context because it allows us to model the number of successes (voters in favor of the initiative) in a fixed number of independent trials (voters). This is particularly useful in situations where we want to calculate the probability of a specific outcome.

Q: Can the probability of exactly 2 voters being in favor of the ballot initiative be affected by other factors?

A: Yes, the probability of exactly 2 voters being in favor of the ballot initiative can be affected by other factors, such as the sample size, the probability of success, and the number of trials. Additionally, demographic factors, such as age or income level, may also impact the probability.

Q: How can the results of this study be applied in real-world scenarios?

A: The results of this study can be applied in real-world scenarios, such as:

• Informing decision-making in elections or referendums
• Developing targeted marketing campaigns to influence public opinion
• Conducting surveys to gauge public opinion on specific issues

Q: What are some potential limitations of this study?

A: Some potential limitations of this study include:

• The sample size is relatively small, which may limit the generalizability of the results
• The study assumes that the probability of a single voter being in favor of the initiative is $30\%$, which may not be accurate in reality
• The study does not account for potential biases and confounding variables in the data

Q: What are some potential future research directions?

A: Some potential future research directions include:

• Investigating the impact of different sample sizes on the probability of exactly 2 voters being in favor of the initiative
• Examining the relationship between the probability of exactly 2 voters being in favor of the initiative and other demographic factors
• Developing more advanced statistical models to account for potential biases and confounding variables in the data

Conclusion

In conclusion, this article addresses some of the most frequently asked questions related to the probability of exactly 2 voters being in favor of the ballot initiative. By understanding the binomial distribution and its application in this context, we can gain valuable insights into public opinion and make informed decisions in real-world scenarios.