a) Find P(X = 4). For this question, I used the formula for expected value E[X] = p(x)*X, and rewrote it ad p(x)*4 = 8.1
This gave me the answer of X = 2.025, however I was told that my answer was wrong and was hoping someone could provide some further clarification on how to solve these types of problems.
b) Find P(X >= 8): Unsure of how to go about this one because of the inequality.
No. The mean and variance of a binomial random variable are, respectively, $$\operatorname{E}[X] = np, \quad \operatorname{Var}[X] = np(1-p),$$ where $n$ is the number of Bernoulli trials and $p$ is the probability of success of each trial. Neither of these is equal to $4$, so I don't understand why you would write something like $4 p(x) = 8.1$, nor do I understand how you can conclude that your answer, which is a probability, could be greater than $1$.
The goal of the exercise is to use the fact that $\operatorname{E}[X] = 8.1$ and $\operatorname{Var}[X] = 3.1$, along with the two formulas above, to solve for the parameters $n$ and $p$ of this distribution. Then use these to compute the probability that $X$ equals $4$ using the probability mass function: $$\Pr[X = 4] = \binom{n}{4} p^4 (1-p)^{n-4}.$$