In a binomial experiment, the probability of obtaining exactly $k$ successes in $n$ trials is given by the binomial probability formula:

$P(k) = \binom{n}{k} \cdot p^k \cdot (1 - p)^{n - k}$

Where:

• $n = 8$ is the number of trials,
• $p = 0.5$ is the probability of success on a single trial,
• $k = 2$ is the number of successes.

First, calculate the binomial coefficient $\binom{n}{k}$, which is defined as:

$\binom{n}{k} = \frac{n!}{k!(n - k)!}$

Let's compute $P(2)$.

$P(2) = \binom{8}{2} \cdot (0.5)^2 \cdot (0.5)^6$

Now, I'll calculate this step by step.The probability $P(2)$ is $0.109$ when rounded to three decimal places.

Would you like more details, or do you have any further questions?

Here are 5 related questions to consider:

1. What is the probability of exactly 3 successes in this binomial experiment?
2. How would the probability change if the number of trials was increased to 10?
3. What is the expected number of successes in this experiment?
4. How do you calculate the variance for a binomial distribution?
5. What is the probability of getting at most 2 successes in this experiment?

Tip: In a binomial distribution, as the number of trials increases, the distribution tends to resemble a normal distribution, especially when $p = 0.5$.