find the probability of exactly two successes in five trials of a binomial experiment in which the probability of success is 50%. p = ?%

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 in a single trial, and $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Identify the values of $n$, $k$, and $p$
Here, $n = 5$, $k=2$, $p = 0.5$ (since $50\%=0.5$), and $1-p=0.5$

## Step3: Calculate the combination $C(n,k)$
$$
\begin{align*}
C(5,2)&=\frac{5!}{2!(5 - 2)!}\
&=\frac{5!}{2!3!}\
&=\frac{5\times4\times3!}{2\times1\times3!}\
&=10
\end{align*}
$$

## Step4: Substitute the values into the binomial probability formula
$$
\begin{align*}
P(X = 2)&=C(5,2)\times(0.5)^{2}\times(0.5)^{5 - 2}\
&=10\times0.25\times0.125\
&=10\times\frac{1}{4}\times\frac{1}{8}\
&=\frac{10}{32}\
& = 0.3125
\end{align*}
$$

## Step5: Convert the probability to a percentage
Multiply the probability by $100$ to get the percentage. $0.3125\times100 = 31.25\%$

# Answer:
$31.25\%$

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the binomial probability formula

Step2: Identify 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: