(a) Let $X$ be a random variable that takes only non-negative values. Prove that for any $a > 0$,
\[P(X \geq a) \leq \frac{E[X]}{a}.\]
This inequality is called Markov’s inequality.

(b) Let $X$ be a random variable with finite mean $\mu$ and variance $\sigma^2$. Prove that for any $a >0$,
\[P\left(|X – \mu| \geq a \right) \leq \frac{\sigma^2}{a^2}.\]
This inequality is called Chebyshev’s inequality.

Let $X\sim \mathcal{N}(\mu, \sigma)$ be a normal random variable with parameter $\mu=6$ and $\sigma^2=4$. Find the following probabilities using the Z-table below.

Let $c$ be a positive real number. Suppose that $X$ is a continuous random variable whose probability density function is given by
\begin{align*}
f(x) = \begin{cases}
\frac{1}{x^3} & \text{ if } x \geq c\\
0 & \text{ if } x < c.
\end{cases}
\end{align*}
(a) Determine the value of $c$.

We have a stick of a unit length. Two points on the stick will be selected randomly (uniformly along the length of the stick) and independently. Then we break the stick at these two points so that we get three pieces of the stick. What is the probability that these three pieces form a triangle?

Let $c$ be a fixed positive number. Let $X$ be a random variable that takes values only between $0$ and $c$. This implies the probability $P(0 \leq X \leq c) = 1$. Then prove the next inequality about the variance $V(X)$.
\[V(X) \leq \frac{c^2}{4}.\]

Alice tossed a fair coin until a head occurred. Then Bob tossed the coin until a head occurred. Suppose that the total number of tosses for Alice and Bob was $7$.

Assuming that each toss is independent of each other, what is the probability that Alice tossed the coin exactly three times?

There are two boxes containing red and blue balls. Let us call the boxes Box A and Box B. Each box contains the same number of red and blue balls. More specifically, Box A has 5 red balls and 5 blue balls. Box B has 20 red balls and 20 blue balls. You choose one box. Then draw two balls randomly from the chosen box without replacement, that is, you will not return the first ball into the box before picking up the second ball.

If you draw two balls with the same color, then you win. Otherwise, you lose. To maximize the chance of winning, which box should you pick?

A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and has no idea how to solve the questions. So the student decided to answer each questions randomly. Thus, for each of 10 questions, the student choose one of the 4 answers randomly and each choice is independent each other.

(1) What is the probability that the student answered correctly only one question among the 10 questions?

(2) Determine the probability that the student passes the course.

(3) What is the expected value of the number of questions the student answered correctly?

(4) Find the variance and standard deviation of the number of questions the student answered correctly.

A random variable $X$ is said to be a Bernoulli random variable if its probability mass function is given by
\begin{align*}
P(X=0) &= 1-p\\
P(X=1) & = p
\end{align*}
for some real number $0 \leq p \leq 1$.

(1) Find the expectation of the Bernoulli random variable $X$ with probability $p$.

Alice and Bob play some game against each other. The probability that Alice wins one game is $p$. Assume that each game is independent.

If Alice wins $n$ games before Bob wins $m$ games, then Alice becomes the champion of the game. What is the probability that Alice becomes the champion.

Consider an infinite series of events of rolling a fair six-sided die. Assume that each event is independent of each other. For each of the below, determine its probability.

(1) At least one die lands on the face 5 in the first $n$ rolls. (2) Exactly $k$ dice land on the face 5 in the first $n \geq k$ rolls. (3) Every die roll results in the face 5.

A sequence of events $\{E_n\}_{n \geq 1}$ is said to be increasing if it satisfies the ascending condition
\[E_1 \subset E_2 \subset \cdots \subset E_n \subset \cdots.\]
Also, a sequence $\{E_n\}_{n \geq 1}$ is called decreasing if it satisfies the descending condition
\[E_1 \supset E_2 \supset \cdots \supset E_n \supset \cdots.\]

When $\{E_n\}_{n \geq 1}$ is an increasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
\[\lim_{n \to \infty} E_n := \bigcup_{n=1}^{\infty} E_n.\]

Also, when $\{E_n\}_{n \geq 1}$ is a decreasing sequence, we define a new event denoted by $\lim_{n \to \infty} E_n$ by
\[\lim_{n \to \infty} E_n := \bigcap_{n=1}^{\infty} E_n.\]

(1) Suppose that $\{E_n\}_{n \geq 1}$ is an increasing sequence of events. Then prove the equality of probabilities
\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right).\]
Hence, the limit and the probability are interchangeable.

(2) Suppose that $\{E_n\}_{n \geq 1}$ is a decreasing sequence of events. Then prove the equality of probabilities
\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right). \]

Consider the network diagram in the figure. The diagram consists of five links and each of them fails to communicate with probability $p$. Answer the following questions about this network.

(1) Determine the probability that there exists at least one path from A to B where every link on the path functions without errors. Express the answer in term of $p$.

(2) Assume that exactly one link has failed. In this case, what is the probability that there is a successful path from A to B, that is, each link on the path has not fail.

Let $A, B$ be events with probabilities $P(A)=2/5$, $P(B)=5/6$, respectively. Find the best lower and upper bound of the probability $P(A \cap B)$ of the intersection $A \cap B$. Namely, find real numbers $a, b$ such that
\[a \leq P(A \cap B) \leq b\]
and $P(A \cap B)$ could take any values between $a$ and $b$.

A researcher conducted the following experiment. Students were grouped into two groups. The students in the first group had more than 6 hours of sleep and took a math exam. The students in the second group had less than 6 hours of sleep and took the same math exam.

The pass rate of the first group was twice as big as the second group. Suppose that $60\%$ of the students were in the first group. What is the probability that a randomly selected student belongs to the first group if the student passed the exam?