assume that a procedure yields a binomial distribution with ( n = 8 ) trials and a probability of success of ( p = 0.30 ). use a binomial probability table to find the probability that the number of successes ( x ) is exactly 4. click on the icon to view the binomial probabilities table. ( p(4)=square ) (round to three decimal places as needed)

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Accepted Answer

# Explanation:
## Step1: Recall binomial probability formula
The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $C(n,k)=\frac{n!}{k!(n - k)!}$, $n = 8$, $k = 4$, $p=0.3$, and $1-p = 0.7$.
## Step2: Calculate combination $C(8,4)$
$$
\begin{align*}
C(8,4)&=\frac{8!}{4!(8 - 4)!}\
&=\frac{8\times7\times6\times5}{4\times3\times2\times1}\
&=70
\end{align*}
$$
## Step3: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.3)^{4}=0.0081$, $(1 - p)^{n - k}=(0.7)^{4}=0.2401$
## Step4: Calculate $P(X = 4)$
$$
\begin{align*}
P(X = 4)&=C(8,4)\times p^{4}\times(0.7)^{4}\
&=70\times0.0081\times0.2401\
&=70\times0.00194481\
&=0.1361367
\end{align*}
$$

# Answer:
$0.136$

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Question

Explanation:

Step1: Recall binomial probability formula

Step2: Calculate combination \(C(8,4)\)

Step3: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step4: Calculate \(P(X = 4)\)

Answer: