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

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

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

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

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

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

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

Prove that for a binomial random variable $X$, the success probability satisfies $p=\frac{E(X)-\text{Var}(X)}{E(X)}.$

In quality control, a batch produces $X$ defective items in $50$ trials, with $E(X)=3$ and $\text{Var}(X)=2.1$. Assuming a binomial model, find the defect probability $p$.

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

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