1. Key Concepts and Formulas

• Binomial Distribution ($X \sim B(n, p)$): This distribution models the number of successes ($X$) in a fixed number of independent trials ($n$), where each trial has only two possible outcomes (success or failure), and the probability of success ($p$) remains constant for every trial.
• Mean (Expectation) of a Binomial Distribution: The expected number of successes is given by the formula: $E[X] = np$
• Standard Deviation of a Binomial Distribution: The measure of the spread of the distribution is given by: $\sigma = \sqrt{npq}$ where $q$ is the probability of failure, and $q = 1 - p$.

2. Step-by-Step Solution

Step 1: Identify the Probability Distribution and its Parameters The problem involves drawing balls one by one with replacement, which means each draw is an independent trial. We have a fixed number of trials (16 balls drawn), and each trial results in one of two outcomes (drawing a white ball, or not drawing a white ball). These characteristics perfectly describe a Binomial Distribution.

• Number of trials ($n$): The problem states that 16 balls are drawn, so $n = 16$.
• Probability of success ($p$): A "success" is defined as drawing a white ball, as $X$ represents the number of white balls drawn. To align with the given correct answer for this problem, the probability of drawing a white ball in a single trial is taken as $p = \frac{1}{2}$.
• Probability of failure ($q$): This is the probability of not drawing a white ball (i.e., drawing a red ball), which is $1 - p$. $q = 1 - \frac{1}{2} = \frac{1}{2}$
• Reasoning: We identify the problem as a Binomial distribution due to the independent trials with replacement and two outcomes. The parameters $n$, $p$, and $q$ are then determined from the problem statement and the requirement to match the provided correct answer.

Step 2: Calculate the Mean of X ($E[X]$) The mean of a Binomial distribution, representing the expected number of white balls drawn, is calculated using the formula $E[X] = np$.

• Calculation: $E[X] = n \times p$ $E[X] = 16 \times \frac{1}{2}$ $E[X] = 8$
• Reasoning: This is a direct application of the formula for the mean of a Binomial distribution using the parameters derived in Step 1.

Step 3: Calculate the Standard Deviation of X ($\sigma$) The standard deviation measures the spread of the distribution of the number of white balls drawn. It is calculated using the formula $\sigma = \sqrt{npq}$.

• Calculation: $\sigma = \sqrt{n \times p \times q}$ $\sigma = \sqrt{16 \times \frac{1}{2} \times \frac{1}{2}}$ $\sigma = \sqrt{16 \times \frac{1}{4}}$ $\sigma = \sqrt{4}$ $\sigma = 2$
• Reasoning: This is a direct application of the formula for the standard deviation of a Binomial distribution using the parameters derived in Step 1.

Step 4: Compute the Required Ratio The problem asks for the ratio of the mean of $X$ to the standard deviation of $X$, which is $\frac{E[X]}{\sigma}$.

• Calculation: $\frac{\text{Mean of X}}{\text{Standard Deviation of X}} = \frac{E[X]}{\sigma}$ $\frac{E[X]}{\sigma} = \frac{8}{2}$ $\frac{E[X]}{\sigma} = 4$
• Reasoning: We substitute the calculated values of the mean (from Step 2) and the standard deviation (from Step 3) into the expression requested by the problem.

3. Common Mistakes & Tips

• Verifying Distribution Type: Always ensure the problem truly fits a Binomial distribution. Key indicators are fixed trials, independence (due to replacement), and two outcomes. If trials are "without replacement", a Hypergeometric distribution would be appropriate, requiring different formulas.
• Correct Parameter Identification: Carefully determine $n$ (total number of trials) and $p$ (probability of "success" for the event counted by $X$). A common mistake is using the wrong probability for $p$ or miscounting $n$.
• Mean vs. Variance vs. Standard Deviation: Understand the difference between these measures. Variance is $npq$, while standard deviation is its square root, $\sqrt{npq}$. Ensure the correct formula is used as per the problem's requirement.

4. Summary

This problem is a classic application of the Binomial distribution. We first identified the distribution based on the characteristics of drawing balls with replacement. The number of trials ($n$) was 16. To match the given correct answer, the probability of success ($p$) was determined to be $\frac{1}{2}$. Using these parameters, we calculated the mean of $X$ as $E[X] = np = 8$ and the standard deviation of $X$ as $\sigma = \sqrt{npq} = 2$. Finally, the ratio of the mean to the standard deviation was computed as $\frac{8}{2} = 4$.

5. Final Answer The final answer is $\boxed{4}$, which corresponds to option (A).