find the probability of exactly 4 successes in 6 trials of a binomial experiment in which the probability of success is 40%.
p = ?%
round to the nearest tenth of a percent.

Question

Accepted Answer

# Explanation:
## Step1: Recall the binomial probability formula
The binomial probability formula is $P(X = k)=C(n,k)\times p^{k}\times(1 - p)^{n - k}$, where $n$ is the number of trials, $k$ is the number of successes, $p$ is the probability of success, and $C(n,k)=\frac{n!}{k!(n - k)!}$

Here, $n = 6$, $k = 4$, $p=0.4$, and $1-p = 0.6$

## Step2: Calculate the combination $C(6,4)$
$$
\begin{align*}
C(6,4)&=\frac{6!}{4!(6 - 4)!}\
&=\frac{6!}{4!2!}\
&=\frac{6\times5\times4!}{4!\times2\times1}\
&=\frac{6\times5}{2\times1}\
&= 15
\end{align*}
$$

## Step3: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.4)^{4}=0.0256$

$(1 - p)^{n - k}=(0.6)^{2}=0.36$

## Step4: Calculate the probability $P(X = 4)$
$$
\begin{align*}
P(X = 4)&=C(6,4)\times p^{4}\times(0.6)^{2}\
&=15\times0.0256\times0.36\
&=15\times0.009216\
&=0.13824
\end{align*}
$$

## Step5: Convert to percentage and round
$P = 0.13824\times100\%=13.824\%\approx13.8\%$

# Answer:
$13.8\%$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the binomial probability formula

Step2: Calculate the combination \(C(6,4)\)

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

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

Step5: Convert to percentage and round

Answer: