Evaluate 

Upper C Subscript n comma x Baseline p Superscript x Baseline q Superscript n minus xCn,xpxqn−x 

for the values of​ n, x, and p given below.

nequals=55​, 

xequals=22​, 

pequals=one half12

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Part 1

Upper C Subscript n comma x Baseline p Superscript x Baseline q Superscript n minus xCn,xpxqn−xequals=enter your response here 

​(Round to three decimal places as​ needed.)




To evaluate the binomial probability formula \(\binom{n}{x} p^x q^{n-x}\) for the given values \( n = 5 \), \( x = 2 \), and \( p = \frac{1}{2} \), we follow these steps:

1. Calculate the binomial coefficient \(\binom{n}{x}\).
2. Calculate \( p^x \).
3. Calculate \( q^{n-x} \), where \( q = 1 - p \).
4. Multiply these values together.

### Step 1: Calculate the Binomial Coefficient

\[
\binom{5}{2} = \frac{5!}{2!(5-2)!} = \frac{5!}{2! \cdot 3!} = \frac{120}{2 \cdot 6} = \frac{120}{12} = 10
\]

### Step 2: Calculate \( p^x \)

\[
p = \frac{1}{2}, \quad x = 2
\]
\[
p^x = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
\]

### Step 3: Calculate \( q^{n-x} \)

\[
q = 1 - p = 1 - \frac{1}{2} = \frac{1}{2}
\]
\[
n - x = 5 - 2 = 3
\]
\[
q^{n-x} = \left(\frac{1}{2}\right)^3 = \frac{1}{8}
\]

### Step 4: Multiply These Values Together

\[
\binom{5}{2} p^2 q^{3} = 10 \cdot \frac{1}{4} \cdot \frac{1}{8} = 10 \cdot \frac{1}{32} = \frac{10}{32} = \frac{5}{16} \approx 0.3125
\]

### Final Answer

\[
\binom{5}{2} \left(\frac{1}{2}\right)^2 \left(\frac{1}{2}\right)^{3} = 0.313
\]

(Rounded to three decimal places)

So, the value is approximately \(0.313\).

Do you have any questions or need further details?

### Related Questions:

1. What is the probability of getting exactly 3 successes with \( n = 5 \) and \( p = \frac{1}{2} \)?
2. How does the binomial coefficient change if \( n \) is increased to 6?
3. What is the probability of getting exactly 0 successes with \( n = 5 \) and \( p = \frac{1}{2} \)?
4. How does the probability change if \( p \) is increased to \( \frac{3}{4} \)?
5. What is the cumulative probability of getting at least 2 successes with \( n = 5 \) and \( p = \frac{1}{2} \)?
6. How do you compute the mean and variance of a binomial distribution?
7. What is the probability of getting at most 1 success with \( n = 5 \) and \( p = \frac{1}{2} \)?
8. How does changing \( x \) affect the overall probability in a binomial distribution?

**Tip:** When dealing with probabilities, ensure that your calculations are precise, and rounding should be done at the final step to maintain accuracy.

Learn how to evaluate the binomial probability formula C(5, 2) * (1/2)^2 * (1/2)^3 with a detailed step-by-step solution. This problem covers binomial coefficients and probabilities, suitable for advanced high school students.

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