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 $\mathrm{P}(\mathrm{X})$ using the normal distribution and compare the result with the exact probability.
For $n=54, p=0.6$, and $X=36$, find $P(X)$.
$P(X)=\square$ (Round to four decimal places as needed.)
Answer
So, the final answer is \(\boxed{0.0687}\).
Step 1 ：Given that the total number of trials, \(n = 54\), the probability of success, \(p = 0.6\), and the number of successes we're interested in, \(X = 36\).
Step 2 ：We need to calculate \(P(X=36)\) 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.
Step 3 ：First, calculate the combination \(C(n, k)\), which is \(C(54, 36) = 96926348578605\).
Step 4 ：Then, calculate the probability \(p^k * (1-p)^{n-k}\), which is \(0.6^{36} * (1-0.6)^{54-36} = 7.088018749850916e-16\).
Step 5 ：Finally, multiply the combination and the probability to get \(P(X=36)\), which is \(96926348578605 * 7.088018749850916e-16 = 0.0687015776079738\).
Step 6 ：Round the result to four decimal places, we get \(P(X=36) = 0.0687\).
Step 7 ：So, the final answer is \(\boxed{0.0687}\).
