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 probabilitv
For $n=54, p=0.6$, and $X=36$, find $P(X)$.
Approximate $P(X)$ using the normal distribution. Select the correct choice below and fill in any answer boxes in your choice.
A. $P(X)=\square$ (Round to four decimal places as needed.)
B. The normal distribution cannot be used.

Step 1 ：Given that the number of trials \(n = 54\), the probability of success \(p = 0.6\), and the number of successes \(X = 36\).

Step 2 ：First, we calculate the exact probability \(P(X)\) using the binomial probability formula: \(P(X=k) = C(n, k) * p^k * (1-p)^{n-k}\), where \(C(n, k)\) is the combination of \(n\) items taken \(k\) at a time, \(p\) is the probability of success, and \(k\) is the number of successes.

Step 3 ：Substituting the given values into the formula, we find that the exact probability \(P(X)\) is approximately 0.0687.

Step 4 ：We then check if the normal distribution can be used to approximate this probability. The conditions for using the normal distribution are that the number of trials \(n\) is large and the probability of success \(p\) is not too close to 0 or 1. In this case, \(n=54\) which is a reasonably large number and \(p=0.6\) is not too close to 0 or 1. Therefore, we can use the normal distribution to approximate \(P(X)\).

Step 5 ：The normal approximation to the binomial distribution is given by: \(Z = \frac{X - np}{\sqrt{np(1-p)}}\), where \(Z\) is the standard normal random variable, \(X\) is the binomial random variable, \(n\) is the number of trials, and \(p\) is the probability of success.

Step 6 ：Substituting the given values into the formula, we find that the approximate probability \(P(X)\) using the normal distribution is approximately 0.8413.

Step 7 ：\(\boxed{\text{Final Answer: } P(X) \approx 0.0687}\)