use ( n = 6 ) and ( p = 0.5 ) to complete parts (a) through (d) below.
(a) construct a binomial probability distribution with the given parameters.
(round to four decimal places as needed.)

Question

Accepted Answer

# Explanation:
## Step1: Recall the binomial probability formula
The binomial probability formula is $P(x)={n\choose x}p^{x}(1 - p)^{n - x}$, where ${n\choose x}=\frac{n!}{x!(n - x)!}$, $n = 6$, $p=0.5$, and $1-p = 0.5$.

## Step2: Calculate $P(0)$
$$
\begin{align*}
{n\choose x}&=\frac{6!}{0!(6-0)!}=1\
P(0)&={6\choose0}(0.5)^{0}(0.5)^{6}\
&=1\times1\times(0.5)^{6}\
&=\frac{1}{64}\approx0.0156
\end{align*}
$$

## Step3: Calculate $P(1)$
$$
\begin{align*}
{n\choose x}&=\frac{6!}{1!(6 - 1)!}=\frac{6!}{1!5!}=6\
P(1)&={6\choose1}(0.5)^{1}(0.5)^{5}\
&=6\times0.5\times(0.5)^{5}\
&=6\times\frac{1}{2}\times\frac{1}{32}=\frac{6}{64}\approx0.0938
\end{align*}
$$

## Step4: Calculate $P(2)$
$$
\begin{align*}
{n\choose x}&=\frac{6!}{2!(6-2)!}=\frac{6\times5}{2\times 1}=15\
P(2)&={6\choose2}(0.5)^{2}(0.5)^{4}\
&=15\times(0.5)^{6}\
&=\frac{15}{64}\approx0.2344
\end{align*}
$$

## Step5: Calculate $P(3)$
$$
\begin{align*}
{n\choose x}&=\frac{6!}{3!(6 - 3)!}=\frac{6\times5\times4}{3\times2\times1}=20\
P(3)&={6\choose3}(0.5)^{3}(0.5)^{3}\
&=20\times(0.5)^{6}\
&=\frac{20}{64}\approx0.3125
\end{align*}
$$

## Step6: Calculate $P(4)$
Since $P(4) = P(2)$ (by symmetry of the binomial distribution when $p = 0.5$), $P(4)\approx0.2344$

## Step7: Calculate $P(5)$
Since $P(5)=P(1)$ (by symmetry of the binomial distribution when $p = 0.5$), $P(5)\approx0.0938$

## Step8: Calculate $P(6)$
Since $P(6)=P(0)$ (by symmetry of the binomial distribution when $p = 0.5$), $P(6)\approx0.0156$

# Answer:
| $x$ | $P(x)$ |
| ---- | ---- |
| $0$ | $0.0156$ |
| $1$ | $0.0938$ |
| $2$ | $0.2344$ |
| $3$ | $0.3125$ |
| $4$ | $0.2344$ |
| $5$ | $0.0938$ |
| $6$ | $0.0156$ |

\(x\)	\(P(x)\)
\(1\)	\(0.0938\)
\(2\)	\(0.2344\)
\(3\)	\(0.3125\)
\(4\)	\(0.2344\)
\(5\)	\(0.0938\)
\(6\)	\(0.0156\)

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the binomial probability formula

Step2: Calculate \(P(0)\)

Step3: Calculate \(P(1)\)

Step4: Calculate \(P(2)\)

Step5: Calculate \(P(3)\)

Step6: Calculate \(P(4)\)

Step7: Calculate \(P(5)\)

Step8: Calculate \(P(6)\)

Answer: