Urn I contains five red chips and four white chips; urn II contains four red and five white chips.Two chips are drawn simultaneously from urn I and placed into urn II. Then a single chip is drawn from urn II. What is the probability that the chip drawn from urn II is white?

In semiconductor manufacturing, wet chemical etching is often used to remove silicon from the backs of wafers prior to metalization. The etch rate is an important characteristic in this process and known to follow a normal distribution. Two different etching solutions have been compared, using two random samples of 10 wafers for etch solution. The observed etch rates are as follows (in mils/min):

$$\begin{matrix} \text{Solution 1} & \quad & \text{Solution 2}\\ \text{9.9} & \text{10.6} & \text{10.2} & \text{10.0}\\ \text{9.4} & \text{10.3} & \text{10.6} & \text{10.2}\\ \text{9.3} & \text{10.0} & \text{10.7} & \text{10.7}\\ \text{9.6} & \text{10.3} & \text{10.4} & \text{10.4}\\ \text{10.2} & \text{10.1} & \text{10.5} & \text{10.3}\\ \end{matrix}$$

(a) Do the data support the claim that the mean etch rate is the same for both solutions? In reaching your conclusions, use a fixed-level test with $\alpha = 0.05$ and assume that both population variances are equal. (b) Calculate a P-value for the test in part (a). (c) Find a 95% CI on the difference in mean etch rates. (d) Construct normal probability plots for the two samples. Do these plots provide support for the assumptions of normality and equal variances? Write a practical interpretation for these plots.

An urn contains four red chips, three white chips, and two blue chips. A random sample of size 3 is drawn without replacement. Let X denote the number of white chips in the sample and Y the number of blue chips. Write a formula for the joint pdf of X and Y.

One of the popular tourist attractions in Alaska is watching black bears catch salmon swimming upstream to spawn. Not all “black” bears are black, though-some are tan-colored. Suppose that six black bears and three tan-colored bears are working the rapids of a salmon stream. Over the course of an hour, six different bears are sighted. What is the probability that those six include at least twice as many black bears as tan-colored bears?

Marcos traveled in his car from Phoenix to Flagstaff, a distance of 155 miles. a. Determine the amount of time required for Marcos to travel from Phoenix to Flagstaff if his average speed for the trip was 68 miles per hour. b. Construct a possible distance-time graph of Marcos's trip from Phoenix to Flagstaff. Be sure to label your axes. c. On the same axes, construct a graph that represents, the distance-time graph that represents another car traveling at a constant speed for the entire trip.

Building permits were issued last year to three contractors starting up a new subdivision: Tara Construction built two houses; Westview, three houses; and Hearthstone, six houses. Tara’s houses have a 60% probability of developing leaky basements; homes built by Westview and Hearthstone have that same problem 50% of the time and 40% of the time, respectively. Yesterday, the Better Business Bureau received a complaint from one of the new homeowners that his basement is leaking. Who is most likely to have been the contractor?

A fair coin is tossed three times. Let X be the number of heads in the tosses minus the number of tails. Find $p_{X}(k)$.

Brett and Margo have each thought about murdering their rich Uncle Basil in hopes of claiming their inheritance a bit early. Hoping to take advantage of Basil’s predilection for immoderate desserts, Brett has put rat poison into the cherries flambe; Margo, unaware of Brett’s activities, has laced the chocolate mousse with cyanide. Given the amounts likely to be eaten, the probability of the rat poison being fatal is 0.60; the cyanide, 0.90. Based on other dinners where Basil was presented with the same dessert options, we can assume that he has a 50% chance of asking for the cherries flambe, a 40% chance of ordering the chocolate mousse, and a 10% chance of skipping dessert altogether. No sooner are the dishes cleared away than Basil drops dead. In the absence of any other evidence, who should be considered the prime suspect?

A display case contains thirty-five gems, of which ten are real diamonds and twenty-five are fake diamonds. A burglar removes four gems at random, one at a time and without replacement. What is the probability that the last gem she steals is the second real diamond in the set of four?

The manager of a fleet of automobiles is testing two brands of radial tires. She assigns one tire of each brand at random to the two rear wheels of eight cars and runs the cars until the tires wear out. The data are shown here (in kilometers). Find a 99% CI on the difference in mean life. Which brand would you prefer, based on this calculation?

$$\begin{matrix} \text{Car} & \text{Brand 1} & \text{Brand 2} & \text{Car} & \text{Brand 1} & \text{Brand 2}\\ \text{1} & \text{36,925} & \text{34,318} & \text{5} & \text{37,210} & \text{38,015}\\ \text{2} & \text{45,300} & \text{42,280} & \text{6} & \text{48,360} & \text{47,800}\\ \text{3} & \text{36,240} & \text{35,500} & \text{7} & \text{38,200} & \text{37,810}\\ \text{4} & \text{32,100} & \text{31,950} & \text{8} & \text{33,500} & \text{33,215}\\ \end{matrix}$$

Urn I contains two white chips and one red chip; urn II has one white chip and two red chips. One chip is drawn at random from urn I and transferred to urn II. Then one chip is drawn from urn II. Suppose that a red chip is selected from urn II. What is the probability that the chip transferred was white?

Country A inadvertently launches ten guided missiles-six armed with nuclear warheads-at Country B. In response, Country B fires seven antiballistic missiles, each of which will destroy exactly one of the incoming rockets. The antiballistic missiles have no way of detecting, though, which of the ten rockets are carrying nuclear warheads. What are the chances that Country B will be hit by at least one nuclear missile?

Doomsday Airlines (“Come Take the Flight of Your Life”) has two dilapidated airplanes, one with two engines, and the other with four. Each plane will land safely only if at least half of its engines are working. Each engine on each aircraft operates independently and each has probability p = 0.4 of failing. Assuming you wish to maximize your survival probability, which plane should you fly on?

The IQs of nine randomly selected people are recorded. Let $\bar{Y}$ denote their average. Assuming the distribution from which the $Y_{i}$'s were drawn is normal with a mean of 100 and a standard deviation of 16, what is the probability that $\bar{Y}$ will exceed 103? What is the probability that any arbitrary $Y_{i}$ will exceed 103? What is the probability that exactly three of the $Y_{i}$'s will exceed 103?