We are tasked with finding the probability $P(X > 4)$, where $X$ is a binomial random variable with parameters $n = 8$ and $p = 0.7$.

The binomial distribution is given by the formula:

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

However, we need $P(X > 4)$, which is the complement of $P(X \leq 4)$. So, we can calculate it as:

$P(X > 4) = 1 - P(X \leq 4)$

To find $P(X \leq 4)$, we sum up the probabilities from $X = 0$ to $X = 4$:

$P(X \leq 4) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)$

Now, let's compute this probability. I'll calculate it step by step.The probability $P(X > 4)$ is approximately $0.8059$ when rounded to four decimal places.

Would you like any further explanation on this, or have additional questions? Here are some related questions you might explore:

1. How is the binomial distribution related to the normal distribution?
2. How would the probability change if $p = 0.5$?
3. What is the expected value of $X$ in this scenario?
4. How would you calculate the variance of $X$?
5. What happens to the probability if we increase the number of trials $n$?

Tip: For large values of $n$ and $p$ not too close to 0 or 1, the binomial distribution can be approximated by a normal distribution for easier calculations.