Current learning ohiective Binomial Distrioution Question to Score: oof \& points In a population where \( 19 \% \) of voters prefer Candidate A, an organization conducts a poll of 9 voters. Find the probability that 4 of the 9 voters will prefer Candidate A. Probability: Enter your answer

To find the probability that exactly 4 out of 9 voters prefer Candidate A (where 19% prefer Candidate A), you can use the binomial probability formula: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] Where: - \( n = 9 \) (the number of trials) - \( k = 4 \) (the number of successful outcomes) - \( p = 0.19 \) (the probability of success) Calculating: 1. \( \binom{9}{4} = \frac{9!}{4!(9-4)!} = 126 \) 2. \( p^k = 0.19^4 \approx 0.001305 \) 3. \( (1-p)^{n-k} = (0.81)^{5} \approx 0.328051 \) Combining these, we have: \[ P(X = 4) = 126 \times 0.001305 \times 0.328051 \approx 0.0528 \] So, the probability that exactly 4 of the 9 voters will prefer Candidate A is approximately **0.0528** or **5.28%**. --- Binomial distribution has roots going back to the 18th century when mathematicians like Jacob Bernoulli started exploring probability. He laid the groundwork for what would come to be known as the Law of Large Numbers, establishing a connection between probability and real-world events, much like how we apply it in polls today! You may have encountered the binomial distribution in action when analyzing sports statistics or in quality control processes. In practical terms, it's like flipping a coin multiple times or predicting how many heads will occur in a series of flips, but transferred to a broader context, such as predicting voter behavior or test outcomes. Understanding how to apply it really amplifies decision-making!