## Probabilities of An Infinite Sequence of Die Rolling

## Problem 745

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.

## Interchangeability of Limits and Probability of Increasing or Decreasing Sequence of Events

## Problem 744

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

\[\lim_{n \to \infty} P(E_n) = P\left(\lim_{n \to \infty} E_n \right). \]

## Linearity of Expectations E(X+Y) = E(X) + E(Y)

## Problem 743

Let $X, Y$ be discrete random variables. Prove the linearity of expectations described as

\[E(X+Y) = E(X) + E(Y).\]

## Successful Probability of a Communication Network Diagram

## Problem 742

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.

## Lower and Upper Bounds of the Probability of the Intersection of Two Events

## Problem 741

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 $b$.

## Find the Conditional Probability About Math Exam Experiment

## Problem 740

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?

Add to solve later## What is the Probability that Selected Coin was Two-Headed?

## Problem 739

There are three coins in a box. The first coin is two-headed. The second one is a fair coin. The third one is a biased coin that comes up heads $75\%$ of the time. When one of the three coins was picked at random from the box and tossed, it landed heads.

What is the probability that the selected coin was the two-headed coin?

Add to solve later## If a Smartphone is Defective, Which Factory Made It?

## Problem 738

A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively. Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively.

If a smartphone of this model is found out to be detective, what is the probability that this smartphone was manufactured in factory C?

Add to solve later## If At Least One of Two Coins Lands Heads, What is the Conditional Probability that the First Coin Lands Heads?

## Problem 737

Two fair coins are tossed. Given that at least one of them lands heads, what is the conditional probability that the first coin lands heads?

Add to solve later## Probability of Having Lung Cancer For Smokers

## Problem 736

Let $C$ be the event that a randomly chosen person has lung cancer. Let $S$ be the event of a person being a smoker.

Suppose that 10% of the population has lung cancer and 20% of the population are smokers. Also, suppose that we know that 70% of all people who have lung cancer are smokers.

Then determine the probability of a person having lung cancer given that the person is a smoker.

Add to solve later## Overall Fraction of Defective Smartphones of Three Factories

## Problem 735

A certain model of smartphone is manufactured by three factories A, B, and C. Factories A, B, and C produce $60\%$, $25\%$, and $15\%$ of the smartphones, respectively.

Suppose that their defective rates are $5\%$, $2\%$, and $7\%$, respectively. Determine the overall fraction of defective smartphones of this model.

Add to solve later## Complement of Independent Events are Independent

## Problem 734

Let $E$ and $F$ be independent events. Let $F^c$ be the complement of $F$.

Prove that $E$ and $F^c$ are independent as well.

Add to solve later## Independent and Dependent Events of Three Coins Tossing

## Problem 733

Suppose that three fair coins are tossed. Let $H_1$ be the event that the first coin lands heads and let $H_2$ be the event that the second coin lands heads. Also, let $E$ be the event that exactly two coins lands heads in a row.

For each pair of these events, determine whether they are independent or not.

Add to solve later## Independent Events of Playing Cards

## Problem 732

A card is chosen randomly from a deck of the standard 52 playing cards.

Let $E$ be the event that the selected card is a king and let $F$ be the event that it is a heart.

Prove or disprove that the events $E$ and $F$ are independent.

Add to solve later## Jewelry Company Quality Test Failure Probability

## Problem 731

A jewelry company requires for its products to pass three tests before they are sold at stores. For gold rings, 90 % passes the first test, 85 % passes the second test, and 80 % passes the third test. If a product fails any test, the product is thrown away and it will not take the subsequent tests. If a gold ring failed to pass one of the tests, what is the probability that it failed the second test?

Add to solve later## What is the Probability that All Coins Land Heads When Four Coins are Tossed If…?

## Problem 730

Four fair coins are tossed.

(1) What is the probability that all coins land heads?

(2) What is the probability that all coins land heads if the first coin is heads?

(3) What is the probability that all coins land heads if at least one coin lands heads?

Add to solve later## Pick Two Balls from a Box, What is the Probability Both are Red?

## Problem 729

There are three blue balls and two red balls in a box.

When we randomly pick two balls out of the box without replacement, what is the probability that both of the balls are red?

Add to solve later## Conditional Probability Problems about Die Rolling

## Problem 728

A fair six-sided die is rolled.

(1) What is the conditional probability that the die lands on a prime number given the die lands on an odd number?

(2) What is the conditional probability that the die lands on 1 given the die lands on a prime number?

Add to solve later## Probability Problems about Two Dice

## Problem 727

Two fair and distinguishable six-sided dice are rolled.

(1) What is the probability that the sum of the upturned faces will equal $5$?

(2) What is the probability that the outcome of the second die is strictly greater than the first die?

Add to solve later## The Number of Elements in a Finite Field is a Power of a Prime Number

## Problem 726

Let $\F$ be a finite field of characteristic $p$.

Prove that the number of elements of $\F$ is $p^n$ for some positive integer $n$.

Add to solve later