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.
\[
\mathrm{n}=50, \mathrm{p}=0.1 \text {, and } \mathrm{X}=15
\]
$P(X)=0.0001$ (Round to four decimal places as needed.)
Can the normal distribution be used to approximate this probability?
A. No, the normal distribution cannot be used because $n p(1-p) \geq 10$.
B. No, the normal distribution cannot be used because $n p(1-p)< 10$.
C. Yes, the normal distribution can be used because $n p(1-p)< 10$.
D. Yes, the normal distribution can be used because $n p(1-p) \geq 10$.
Answer
Final Answer: The exact probability is approximately \(\boxed{0.000056}\). The normal distribution \textbf{cannot} be used to approximate this probability because \(\boxed{np(1-p) < 10}\). Therefore, the correct answer is \textbf{B}. No, the normal distribution cannot be used because \(np(1-p)<10\).
Step 1 ：Given that n=50, p=0.1, and X=15, we first need to calculate the exact probability using the binomial probability formula. The binomial probability formula is given by: \(P(X=k) = C(n, k) * (p^k) * ((1-p)^{(n-k)})\), where n is the number of trials, k is the number of successful trials, p is the probability of success on each trial, and C(n, k) is the binomial coefficient, which can be calculated as \(n! / (k!(n-k)!)\).
Step 2 ：Substituting the given values into the formula, we get \(P(X=15)\).
Step 3 ：The exact probability \(P(X=15)\) is approximately \(0.000056\), which is close to the given value of \(0.0001\).
Step 4 ：Next, we need to check whether the normal distribution can be used to approximate this probability. The rule of thumb is that the normal distribution can be used if \(np(1-p) >= 10\). So we need to calculate \(np(1-p)\) and check whether it's greater than or equal to 10.
Step 5 ：The value of \(np(1-p)\) is 4.5, which is less than 10.
Step 6 ：Therefore, the normal distribution cannot be used to approximate this probability.
Step 7 ：Final Answer: The exact probability is approximately \(\boxed{0.000056}\). The normal distribution \textbf{cannot} be used to approximate this probability because \(\boxed{np(1-p) < 10}\). Therefore, the correct answer is \textbf{B}. No, the normal distribution cannot be used because \(np(1-p)<10\).
