Question Type 3: Using n and p as parameters and finding what values of either satisfy some conditions Exercises

Prove that for a binomial random variable $X$, the success probability $p$ satisfies

Given a binomial random variable $X$ with $E(X) = 5$ and $\text{Var}(X) = 4$, find the probability of success $p$.

For a binomial random variable $X$, it is known that $E(X) = 14$ and $\text{Var}(X) = 8.4$. Find $p$ and $n$.

For a binomial model $X \sim \text{B}(n, p)$ with $\text{E}(X) = 12$ and $\text{Var}(X) = 9$, find the values of $p$ and $n$, and calculate $P(X = 0)$.

Suppose $X \sim \text{Bin}(n, p)$ has $\text{E}(X) = 20$ and $\text{Var}(X) = 16$. Determine $n$ and $p$.

A gamer has a success probability of $0.25$ each attempt. Let $X$ be the number of successes in $n$ trials. If $\text{Var}(X) = 4.6875$, find $n$ and $E(X)$.

If for a binomial distribution the expected value is twice the variance, $E(X) = 2\text{Var}(X)$, find the value of $p$.

A binomial random variable $X$ has $p=0.4$ and $E(X)=8$. Determine $n$ and $\text{Var}(X)$.

In quality control, the number of defective items $X$ in a batch follows a binomial distribution. It is given that $E(X) = 3$ and $\text{Var}(X) = 2.1$. Find the defect probability $p$.

A binomial random variable $X$ satisfies $E(X)=12$ and $\text{Var}(X)=6$. Determine $n$ and $p$.

If $X \sim \text{Bin}(n, p)$ satisfies $E(X) = 0.6n$ and $\text{Var}(X) = 0.24n$, show that $p = 0.6$.