find the probability of exactly four successes in five trials of a binomial experiment in which the probability of success is 40%. p = ?%

Question

Accepted Answer

# Explanation:
## Step1: Identify 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 $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Determine the values of $n$, $k$, and $p$
Given $n = 5$, $k = 4$, $p=0.4$, then $1 - p = 0.6$

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(5,4)&=\frac{5!}{4!(5 - 4)!}\
&=\frac{5!}{4!1!}\
&=\frac{5\times4!}{4!×1}\
&=5
\end{align*}
$$

## Step4: Substitute the values into the binomial probability formula
$$
\begin{align*}
P(X = 4)&=C(5,4)\times(0.4)^{4}\times(0.6)^{5 - 4}\
&=5\times0.4^{4}\times0.6^{1}\
&=5\times0.0256\times0.6\
&=5\times0.01536\
&=0.0768
\end{align*}
$$

## Step5: Convert the probability to a percentage
$P = 0.0768\times100\%=7.68\%\approx7.7\%$

# Answer:
$7.7\%$

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Identify the binomial probability formula

Step2: Determine the values of \(n\), \(k\), and \(p\)

Step3: Calculate the combination \(C(n,k)\)

Step4: Substitute the values into the binomial probability formula

Step5: Convert the probability to a percentage

Answer: