## 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## The Zero is the only Nilpotent Element of the Quotient Ring by its Nilradical

## Problem 725

Prove that if $R$ is a commutative ring and $\frakN(R)$ is its nilradical, then the zero is the only nilpotent element of $R/\frakN(R)$. That is, show that $\frakN(R/\frakN(R))=0$.

Add to solve later## Three Equivalent Conditions for an Ideal is Prime in a PID

## Problem 724

Let $R$ be a principal ideal domain. Let $a\in R$ be a nonzero, non-unit element. Show that the following are equivalent.

(1) The ideal $(a)$ generated by $a$ is maximal.

(2) The ideal $(a)$ is prime.

(3) The element $a$ is irreducible.

## Every Prime Ideal of a Finite Commutative Ring is Maximal

## Problem 723

Let $R$ be a finite commutative ring with identity $1$. Prove that every prime ideal of $R$ is a maximal ideal of $R$.

Add to solve later## If the Nullity of a Linear Transformation is Zero, then Linearly Independent Vectors are Mapped to Linearly Independent Vectors

## Problem 722

Let $T: \R^n \to \R^m$ be a linear transformation.

Suppose that the nullity of $T$ is zero.

If $\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a linearly independent subset of $\R^n$, then show that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$.

Add to solve later## Find All Values of $x$ such that the Matrix is Invertible

## Problem 721

Given any constants $a,b,c$ where $a\neq 0$, find all values of $x$ such that the matrix $A$ is invertible if

\[

A=

\begin{bmatrix}

1 & 0 & c \\

0 & a & -b \\

-1/a & x & x^{2}

\end{bmatrix}

.

\]

## Find All Eigenvalues and Corresponding Eigenvectors for the $3\times 3$ matrix

## Problem 720

Find all eigenvalues and corresponding eigenvectors for the matrix $A$ if

\[

A=

\begin{bmatrix}

2 & -3 & 0 \\

2 & -5 & 0 \\

0 & 0 & 3

\end{bmatrix}

.

\]

## Find All Values of $a$ which Will Guarantee that $A$ Has Eigenvalues 0, 3, and -3.

## Problem 719

Let $A$ be the matrix given by

\[

A=

\begin{bmatrix}

-2 & 0 & 1 \\

-5 & 3 & a \\

4 & -2 & -1

\end{bmatrix}

\]
for some variable $a$. Find all values of $a$ which will guarantee that $A$ has eigenvalues $0$, $3$, and $-3$.

## Compute the Determinant of a Magic Square

## Problem 718

Let

\[

A=

\begin{bmatrix}

8 & 1 & 6 \\

3 & 5 & 7 \\

4 & 9 & 2

\end{bmatrix}

.

\]
Notice that $A$ contains every integer from $1$ to $9$ and that the sums of each row, column, and diagonal of $A$ are equal. Such a grid is sometimes called a magic square.

Compute the determinant of $A$.

Add to solve later## Are These Linear Transformations?

## Problem 717

Define two functions $T:\R^{2}\to\R^{2}$ and $S:\R^{2}\to\R^{2}$ by

\[

T\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

2x+y \\ 0

\end{bmatrix}

,\;

S\left(

\begin{bmatrix}

x \\ y

\end{bmatrix}

\right)

=

\begin{bmatrix}

x+y \\ xy

\end{bmatrix}

.

\]
Determine whether $T$, $S$, and the composite $S\circ T$ are linear transformations.

## Using Gram-Schmidt Orthogonalization, Find an Orthogonal Basis for the Span

## Problem 716

Using Gram-Schmidt orthogonalization, find an orthogonal basis for the span of the vectors $\mathbf{w}_{1},\mathbf{w}_{2}\in\R^{3}$ if

\[

\mathbf{w}_{1}

=

\begin{bmatrix}

1 \\ 0 \\ 3

\end{bmatrix}

,\quad

\mathbf{w}_{2}

=

\begin{bmatrix}

2 \\ -1 \\ 0

\end{bmatrix}

.

\]