More poker hands. a. What is the probability that a poker hand is a three of a kind? (A three of a kind has three cards of the same value and two other cards of different values, such as three 10s, a 7, and a jack.) b. What is the probability that a poker hand is a full house? (A full house has three cards with one common value and two other cards of another common value, such as three queens and two 4s.) c. What is the probability that a poker hand has one pair? (One pair means two cards have the same value and three other cards have three other values, such as two 9s, a king, an 8, and a 5.) d. What is the probability that a poker hand has two pairs? (Two pairs means two cards have one common value, two more cards have another common value, and a fifth card has yet another value, such as two jacks, two 8s, and a 3.)

The exercise concerns a variant of the Monty Hall problem, in which a contestant choose one of four doors; behind one of the doors is a car, and behind the other three are goats. After the contestant chooses a door, the host opens one of the other three doors that hides a goat. The host then gives the contestant the option of abandoning the chosen door in favor of one of the two still-closed, unchosen doors. For each strategy, what is the probability of winning the car? Switch to one of the unchosen, unopened doors. The choice between the two unchosen, unopened doors is made randomly.

This week at swim practice, Jamie swam a total of 1,500 m, while Ellie swam 1.6 km. How many kilometers did Jamie swim?

$$\begin{array} { l } { \text { Show that the prime subfield of a field of characteristic } p \text { is ring- } } \\ { \text { isomorphic to } Z _ { p } \text { and that the prime subfield of a field of charac- } } \end{array}\text { teristic } 0 \text { is ring-isomorphic to } Q$$

Compute the linear approximation of the function at the given point. $f(w, x, y, z)=w^{2} x y-e^{w y z}$ at (0, 1, -1, 2)

A 5.0-kg bucket of water is raised from a well by a rope. If the upward acceleration of the bucket is 3.0 m/s$^2$, find the force exerted by the rope on the bucket.

A bag contains ten identically wrapped boxes, but the contents of the boxes have different values (e.g. each contains a different amount of money). Alice and Bob are each going to pick one box from the bag. Suppose Alice picks first (one of the ten boxes at random) and then Bob picks at random from the remaining boxes. What is the probability that the contents of Alice's box are more valuable than the contents of Bob's box? Is there an advantage to going first?

Hydrogen is contained in a piston–cylinder device at 14.7 psia and 15 ft^3. At this state, a linear spring (F ~ x) with a spring constant of 15,000 lbf/ft is touching the piston but exerts no force on it. The cross-sectional area of the piston is 3 ft^2. Heat is transferred to the hydrogen, causing it to expand until its volume doubles. Determine (a) the final pressure, (b) the total work done by the hydrogen, and (c) the fraction of this work done against the spring. Also, show the process on a P-V diagram.

Reflect $\overline{AB}$ with endpoints $A(1,2)$ and $B(3,3)$ over the line $x=4$. Now reflect the image over the line $y=-1$. Write the coordinates of the endpoints of the image of $\overline{AB}$ after these two reflections. Describe a single transformation that maps $\overline{AB}$ on to the final image.

Fill in the blank. a. _____

$$\left( \frac { x } { 54 } - \frac { 1 } { 42 } \right) = 7 x - 9$$

b. _____

$$\left( \frac { x } { 54 } - \frac { 1 } { 42 } \right) = 7 x - 9$$

A member of a car-sharing program can use a car for $6 per hour and$.50 per mile. The member uses the car for one day and is charged $44. This situation is given by 6t + 0.5d = 44 where t is the time (in hours) the car is used and d is the distance (in miles) the car is driven. Give three examples of the number of hours the member could have used the car and the number of miles the member could have driven the car.

Prove that, for $2\times 2$ matrix

$$\begin{bmatrix}2&-1\\4&6\end{bmatrix}$$

with a double eigenvalue but a single eigenvector $\overrightarrow{\textbf{v}}=[v_1,v_2],v_2\ne 0$, the matrix

$$\textbf{Q}=\begin{bmatrix}v_1&1\\v_2&0\end{bmatrix}$$

and its inverse $\textbf{Q}^{-1}$ can provide a change of basis for A, such that $\textbf{Q}^{-1}\textbf{AQ}$ is a triangular matrix.

A 3-in-diameter horizontal water jet having a velocity of 90 ft/s strikes a curved plate, which deflects the water $180^{\circ}$ at the same speed. Ignoring the frictional effects, determine the force required to hold the plate against the water stream.

A fair die is rolled 15 times. Calculate the probability that there are: $\mathbf{(a)}$ Exactly three 6s and three 5s $\mathbf{(b)}$ Exactly three 6s, three 5s, and four 4s $\mathbf{(c)}$ Exactly two 6s What is the expected number of 6s obtained?