Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places.

P(X>4)
, n=8
, p=0.7

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.

Assume the random variable X has a binomial distribution with the given probability of obtaining a success. Find the following probability, given the number of trials and the probability of obtaining a success. Round your answer to four decimal places. P(X > 4), n=8, p=0.7.

Find the probability P(X > 4) for a binomial distribution with n=8 trials and a success probability of p=0.7. This solution covers core probability concepts, including the binomial theorem and related formulas. Perfect for students in Grades 10-12.

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