Compute $\mathrm{P}(\mathrm{X})$ 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.
\[
\mathrm{n}=50, \mathrm{p}=0.1, \text { and } \mathrm{X}=15
\]
$P(X)=\square$ (Round to four decimal places as needed.)

Step 1 ：Given that n = 50, p = 0.1, and X = 15, we can use the binomial probability formula to calculate P(X=15). The formula is given by: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))\), where n is the number of trials, p is the probability of success on each trial, k is the number of successes we're interested in, and C(n, k) is the number of combinations of n items taken k at a time.

Step 2 ：Using the formula, we find that the exact probability P(X=15) is approximately 0.000056.

Step 3 ：We can check if the normal distribution can be used to approximate this probability. The rule of thumb is that the normal approximation is reasonable if both np and n(1-p) are greater than 5. In this case, both conditions are met.

Step 4 ：We can use the normal distribution to approximate P(X=15). The normal approximation to the binomial distribution is given by: \(Z = \frac{X - np}{\sqrt{np(1-p)}}\), where Z is a standard normal random variable.

Step 5 ：Using the standard normal distribution table, we find that the approximate probability P(X=15) using the normal distribution is also approximately 0.000056.

Step 6 ：Comparing the exact probability with the approximate probability, we find that they are approximately equal.

Step 7 ：Final Answer: The exact probability \(P(X=15)\) is approximately \(\boxed{0.000056}\) and the approximate probability using the normal distribution is also approximately \(\boxed{0.000056}\).