Question Type 1: Finding the probability of certain outcomes of random variables Bootcamps

A fair six‐sided die is rolled once. Let $X$ be the outcome. Find the probability that $X$ is even or greater than 4.

A random variable $X$ has pmf $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.3$, $P(X=3)=0.4$. Find $P(X\le2)$.

Two fair dice are rolled once. Let $S$ be their sum. Calculate $P(S\ge9)$.

A biased coin has $P(\text{heads})=0.6$. It is flipped twice. Let $X$ be the number of heads. Find $P(X=1)$.

Let $X\sim\text{Uniform}(0,3)$ be continuous. Calculate $P(1<X<2)$.

Let $X$ be a discrete random variable with probability mass function $P(X=x)=\frac{6}{11x}$ for $x\in\{1,2,3\}$. Calculate $P(X>1)$.

A random variable $X$ takes values $0,1,2,3,4$ with $P(X=0)=0.2,P(X=1)=0.3,P(X=2)=0.25,P(X=3)=0.15,P(X=4)=0.1$. Compute $P(1<X\le4)$.

The pmf of $X$ is $P(X=0)=0.5$, $P(X=1)=0.3$, $P(X=2)=0.2$. Find $P(X=2\mid X>0)$.

Let $X$ be geometric with $P(X=k)=0.3\,(0.7)^{k-1}$ for $k=1,2,\dots$. Find $P(X>3)$.

Let $X\sim\text{Binomial}(n=5,p=0.4)$. Determine $P(X\le2)$.

Let $X$ be a discrete variable with $P(X=k)=c\,k^2$ for $k=1,2,3$. (a) Find $c$. (b) Then calculate $P(X=3)$.

Suppose $X\sim\text{Exponential}(\lambda=2)$. Compute $P(X>1)$.