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.