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 further insight into this data, 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 explore the concept of binomial distribution to find the answer.

Understanding the Binomial Distribution

The binomial distribution is a discrete 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 scenario, we have 5 voters, and each voter can either be in favor of the ballot initiative (success) or not (failure). The probability of success, denoted by $p$, is $30\%$ or $0.3$. The probability of failure, denoted by $q$, is $1 - p = 0.7$. We are interested in finding the probability of exactly 2 successes (voters in favor) in 5 trials.

Calculating the Probability

To calculate the probability of exactly 2 successes in 5 trials, we can use the binomial probability formula:

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

where $n$ is the number of trials (5), $k$ is the number of successes (2), $p$ is the probability of success (0.3), and $q$ is the probability of failure (0.7).

Applying the Binomial Probability Formula

Using the formula, we can calculate the probability of exactly 2 successes in 5 trials:

$$P(X = 2) = \binom{5}{2} (0.3)^2 (0.7)^3$$

Calculating the Binomial Coefficient

The binomial coefficient $\binom{5}{2}$ can be calculated as:

$$\binom{5}{2} = \frac{5!}{2! (5-2)!} = \frac{5!}{2! 3!} = \frac{5 \times 4 \times 3!}{2 \times 1 \times 3!} = \frac{5 \times 4}{2 \times 1} = 10$$

Substituting the Binomial Coefficient

Substituting the binomial coefficient into the formula, we get:

$$P(X = 2) = 10 (0.3)^2 (0.7)^3$$

Evaluating the Expression

Evaluating the expression, we get:

$$P(X = 2) = 10 (0.09) (0.343) = 0.3077$$

Rounding the Answer

Rounding the answer to the nearest thousandth, we get:

$$P(X = 2) = 0.308$$

Conclusion

In conclusion, the probability that exactly 2 voters will be in favor of the ballot initiative is approximately $0.308$. This means that if Hans randomly surveys 5 voters, there is a $30.8\%$ chance that exactly 2 voters will be in favor of the ballot initiative.

Discussion

The binomial distribution is a powerful tool for modeling the number of successes in a fixed number of independent trials. In this scenario, we used the binomial probability formula to calculate the probability of exactly 2 successes in 5 trials. The result shows that the probability of exactly 2 voters being in favor of the ballot initiative is approximately $0.308$. This can be useful for Hans in understanding the likelihood of certain outcomes in his survey.

Limitations

One limitation of this analysis is that it assumes that the voters are independent and identically distributed. In reality, the voters may have different characteristics and behaviors that can affect the outcome of the survey. Additionally, the poll may have been conducted under different conditions, such as different sampling methods or different question wording, which can affect the results.

Future Research

Future research can explore the use of more advanced statistical models, such as the generalized linear mixed model, to account for the potential correlations between the voters. Additionally, researchers can investigate the effect of different sampling methods and question wording on the results of the poll.

References

• [1] Johnson, N. L., Kotz, S., & Balakrishnan, N. (1995). Continuous univariate distributions. Vol. 1. Wiley.
• [2] Agresti, A. (2002). Categorical data analysis. Wiley.
• [3] Gelman, A., & Hill, J. (2007). Data analysis using regression and multilevel/hierarchical models. Cambridge University Press.
  

Introduction

In our previous article, we explored the concept of binomial distribution and calculated the probability of exactly 2 voters being in favor of the ballot initiative. In this article, we will address some of the most frequently asked questions related to this topic.

Q: What is the binomial distribution?

A: The binomial distribution is a discrete probability distribution that models the number of successes in a fixed number of independent trials, where each trial has a constant probability of success.

Q: What is the probability of success (p) in this scenario?

A: The probability of success (p) is 30% or 0.3, which represents the proportion of voters who support the ballot initiative.

Q: What is the probability of failure (q) in this scenario?

A: The probability of failure (q) is 1 - p = 0.7, which represents the proportion of voters who do not support the ballot initiative.

Q: How do we calculate the probability of exactly 2 successes in 5 trials?

A: We use the binomial probability formula: P(X = k) = (n choose k) p^k q^(n-k), where n is the number of trials (5), k is the number of successes (2), p is the probability of success (0.3), and q is the probability of failure (0.7).

Q: What is the binomial coefficient (n choose k) in this scenario?

A: The binomial coefficient (n choose k) is 10, which is calculated as 5! / (2! 3!).

Q: How do we evaluate the expression P(X = 2) = 10 (0.3)^2 (0.7)^3?

A: We evaluate the expression by multiplying the binomial coefficient (10) by the probability of success (0.3)^2 and the probability of failure (0.7)^3.

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

A: The final probability is approximately 0.308, which means that there is a 30.8% chance that exactly 2 voters will be in favor of the ballot initiative.

Q: What are some limitations of this analysis?

A: Some limitations of this analysis include the assumption of independent and identically distributed voters, and the potential effects of different sampling methods and question wording on the results.

Q: What are some potential future research directions?

A: Some potential future research directions include using more advanced statistical models, such as the generalized linear mixed model, to account for potential correlations between voters, and investigating the effects of different sampling methods and question wording on the results.

Q: What are some real-world applications of this analysis?

A: This analysis has real-world applications in fields such as politics, marketing, and social sciences, where understanding the probability of certain outcomes is crucial for decision-making and policy development.

Q: How can I apply this analysis to my own research or work?

A: You can apply this analysis to your own research or work by using the binomial probability formula to calculate the probability of certain outcomes, and by considering the limitations and potential future research directions outlined in this article.

Conclusion

In conclusion, the binomial distribution is a powerful tool for modeling the number of successes in a fixed number of independent trials. By understanding the probability of exactly 2 voters being in favor of the ballot initiative, we can gain insight into the likelihood of certain outcomes and make more informed decisions. We hope that this article has been helpful in addressing some of the most frequently asked questions related to this topic.