Question Type 1: Finding the probability of outcomes for binomial distribution using the GDC Exercises

Suppose $X \sim B(30, p)$ and $E(X) = 18$. Find $p$.

For $X \sim B(100, 0.8)$, calculate $P(X > 65)$.

Given $X \sim \text{B}(20, p)$ with variance $5$, find $p$.

A lightbulb manufacturer has a defect rate of $8\%$. In a batch of $200$ bulbs, let $X$ be the number of nondefective bulbs. Find $P(X \ge 190)$.

Let $X \sim B(n, p)$ and suppose its standard deviation equals half its mean. Find a relation between $n$ and $p$.

A student guesses randomly on 15 true/false questions ($p=0.5$). If $X$ is the number of correct answers, find $P(X ≥ 12)$.

Let $X \sim B(20, 0.3)$. Find $P(X \ge 5)$.

For $X \sim \text{B}(80, 0.7)$, calculate the expected value and the standard deviation of $X$.

Let $X \sim B(n, p)$ satisfy $E(X) = 10$ and $\operatorname{Var}(X) = 6$. Find $n$ and $p$.

In 15 independent transmissions, the probability of success is $0.2$. If $X$ is the number of successful transmissions, find $E(X)$ and $\operatorname{Var}(X)$.

For $X \sim B(n, p)$, determine $p$ if the mean is twice the variance.

If $X \sim \text{B}(50, 0.6)$, find the expected value $\text{E}(X)$ and the variance $\text{Var}(X)$.