find the probability of exactly three successes in six 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, and $C(n,k)=\frac{n!}{k!(n - k)!}$

## Step2: Identify the values of $n$, $k$, and $p$
Here, $n = 6$, $k=3$, $p = 0.5$

## Step3: Calculate the combination $C(6,3)$
$$
\begin{align*}
C(6,3)&=\frac{6!}{3!(6 - 3)!}\
&=\frac{6!}{3!3!}\
&=\frac{6\times5\times4\times3!}{3!\times3\times2\times 1}\
&=20
\end{align*}
$$

## Step4: Calculate $p^{k}$ and $(1 - p)^{n - k}$
$p^{k}=(0.5)^{3}=0.125$, $(1 - p)^{n - k}=(0.5)^{3}=0.125$

## Step5: Calculate the probability $P(X = 3)$
$$
\begin{align*}
P(X = 3)&=C(6,3)\times p^{3}\times(1 - p)^{3}\
&=20\times0.125\times0.125\
&=20\times\frac{1}{8}\times\frac{1}{8}\
&=\frac{20}{64}\
& = 0.3125
\end{align*}
$$

## Step6: Convert the probability to a percentage
$P(X = 3)=0.3125\times100\% = 31.25\%\approx31.3\%$

# Answer:
$31.3\%$

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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 \(C(6,3)\)

Step4: Calculate \(p^{k}\) and \((1 - p)^{n - k}\)

Step5: Calculate the probability \(P(X = 3)\)

Step6: Convert the probability to a percentage

Answer: