Compute $P(X)$ using the binomial probability formula. Then determine whether the normal distribution can be used to estimate this probability. If so, approximate $P(X)$ using the normal distribution and compare the result with the exact probability.
\[
n=50, p=0.1, \text { and } X=15
\]
Approximate $\mathrm{P}(\mathrm{X})$ using the normal distribution. Use a standard normal distribution table. Select the correct choice below and fill in any answer boxes in your choice.
A. $P(X)=$
(Round to four decimal places as needed.)
B. There is no solution.

Step 1 ：First, we need to calculate the mean and standard deviation of the binomial distribution. The mean (expected value) of a binomial distribution is given by \(\mu = np\), and the standard deviation is given by \(\sigma = \sqrt{np(1-p)}\).

Step 2 ：Substitute \(n = 50\) and \(p = 0.1\) into the formulas. We get \(\mu = 50 \times 0.1 = 5\) and \(\sigma = \sqrt{50 \times 0.1 \times (1-0.1)} = \sqrt{4.5} \approx 2.12\).

Step 3 ：Next, we need to standardize the random variable \(X = 15\) using the formula \(Z = \frac{X - \mu}{\sigma}\).

Step 4 ：Substitute \(X = 15\), \(\mu = 5\), and \(\sigma = 2.12\) into the formula. We get \(Z = \frac{15 - 5}{2.12} \approx 4.72\).

Step 5 ：Now, we can use the standard normal distribution table to find the probability \(P(Z < 4.72)\). However, the standard normal distribution table usually does not provide values for \(Z > 3.49\), because the probabilities are very close to 1.

Step 6 ：Therefore, we can say that \(P(Z < 4.72) \approx 1\).

Step 7 ：Finally, because \(X = 15\) is greater than the mean \(\mu = 5\), we need to subtract this probability from 1 to get the probability \(P(X = 15)\).

Step 8 ：So, \(P(X = 15) = 1 - P(Z < 4.72) = 1 - 1 = 0\).

Step 9 ：Therefore, the probability that \(X = 15\) is approximately 0 when using the normal distribution to approximate the binomial distribution.