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

Let $X \sim \text{U}(0,3)$. 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 $X$ has probability mass function $P(X=0)=0.1$, $P(X=1)=0.2$, $P(X=2)=0.3$, $P(X=3)=0.4$. 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 $P(X=1)$.

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 $X$ 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 $P(1 < X \le 4)$.

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 $X \sim \text{B}(5, 0.4)$. Determine $P(X \le 2)$.

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

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