Nine West sells women's flip flops in eight different colors. Despite
the available selection, 50% of all filp flops purchased are white.
Suppose 30 buyers are selected at random.
a. Find the mean, variance, and standard deviation of the number of
buyers who purchase white flip flops.
b. Find the probability that the number of white flip flops purchased
will be within two standard deviations of the mean. Compare this
with the predicted result from Chebyshev's Rule.
c. Suppose two groups of 30 customers are independently selected.
What is the probability of at least one group having exactly 15
people who buy white flip flops.

1. What two characteristics of the data do you need to determine which binomial distribution to use?
2. What two characteristics of the data do you need to determine which Poisson distribution to use?
3. What two characteristics of the data do you need to determine which normal distribution to use?

(2) For the following scenarios determine if you should use Binomial, Poisson or Normal Distributions

(a) The experiment/problem involves 2 independent outcomes, a fixed number of trials with a known (fixed) probability of “success”

(b) The experiment/problem involves number of outcomes per area or time

(c) The experiment/problem involves determining the probability of certain measurements

The average number of miles (in thousands) that a car's tire will function before needing replacement is 66 and the standard deviation is 14. Suppose that 18 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of XX? XX ~ N(,)
2. What is the distribution of ¯xx¯? ¯xx¯ ~ N(,)
3. If a randomly selected individual tire is tested, find the probability that the number of miles (in thousands) before it will need replacement is between 69.1 and 73.
4. For the 18 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 69.1 and 73.
5. For part d), is the assumption that the distribution is normal necessary? YesNo

A project has five activities: A, B, C, and D, which must be carried out sequentially. The probability distributions of the number of weeks required to complete each of the activities A, B, and C are uniform in intervals [1,5], [2,3], and [3,6], respectively. The number of weeks required to complete activity D is a beta distribution with shape parameters a=2 and b=5, lower bound 2, and upper bound 10. Use simulation to forecast the total completion time of the project.

a. What are the mean and standard deviation of the total completion time?

b. What are the upper and lower bounds of a 95% confidence interval for the completion time?

c. What is the probability that the project will be completed within

The average number of miles (in thousands) that a car's tire will function before needing replacement is 66 and the standard deviation is 11. Suppose that 43 randomly selected tires are tested. Round all answers to 4 decimal places where possible and assume a normal distribution.

1. What is the distribution of X? X ~ N
2. What is the distribution of ¯x? ¯x ~ N
3. If a randomly selected individual tire is tested, find the probability that the number of miles (in thousands) before it will need replacement is between 65.6 and 66.7.
4. For the 43 tires tested, find the probability that the average miles (in thousands) before need of replacement is between 65.6 and 66.7.
5. For part d), is the assumption that the distribution is normal necessary?  

The mean number of minutes for app engagement by a table user is 7.5 minutes. Suppose the standard deviation is 1.25 minutes. Take a sample of 80.

a. What are the mean and standard deviation for the sample mean number of app engagement by a tablet user?

b. Find the 90th percentile for the sample mean time for app engagement for a tablet user. Interpret this value in a complete sentence.

c. Find the probability that the sample mean is between 7 and 7.75 minutes.

d. What are the mean and standard deviation for the sums?

e. Find the 90th percentile for the sum of the sample. Interpret this value in a complete sentence.

f. Find the probability that the sum of the sample is at least 10 hours.

The formal study of probability began with questions regarding gambling and games of chance. The conventional analysis of gambling is based on the expected values of these games which is always negative for the player and positive for the casino house. The absolute values of the two are exactly the same. Therefore, what the player loses equals what the house wins (in the long run). If the expected value of a game for the player is 0, then the game is 'fair'. Note that fair games would earn zero revenue for the casino, so casinos cannot afford to provide players with fair games! To earn revenue for the casino, games must be 'unfair', to the advantage of the house. The 'unfairness' of casino games is well-known to players. The players, however, knowingly play the 'unfair' games!

(Reference: http://www.casinosprofit.com/the-expected-value-of.html )

Consider the game of roulette, a well-known casino game. Originating in late seventeenth-century France, this game is typically played on a wheel with 38 slots numbered 00, 0, and 1 through 36, although not in sequence. The 00 and 0 slots are green, and all other slots alternate in color, black/red/black (and so on), enabling players to place wagers many different ways. The wheel is spun, then a ball is dropped onto the wheel and is equally likely to end up in any one of the 38 slots.

There are many ways to bet and the payoffs are different for different wagers. For example, to make a "straight" bet (payoff 35:1), the chip(s) will be placed in one of the numbered spaces on the game board, and if the ball ends up in that slot, the player wins $35 for every $1 wagered. Note that the game of roulette returns your initial bet to you if you win, so with this straight bet, a player who bets $1 will either have a gain of $35 or a loss of $1.

a) Marco decides to play roulette for the rest of the evening and repeatedly places a $1 wager on the number 22. What is the expected value of this game? (In other words, what is his expected net gain over many, many repeated plays?) Explain why this is an 'unfair' game.

b) Maxine is a little less adventurous and hopes to win more often (and lose less often) so she repeatedly places her $1 bet on red (which has more ways to win but a winning payoff of only 1:1, $1 won for every $1 bet). Should she expect to break even by playing this way since the payoff is 1:1? Does she have a 50/50 chance of winning each time the wheel is spun? What is the expected net gain? Explain.

c) Recall last week's discussion on The Law of Averages vs. The Law of Large Numbers and combine that with the questions that you just answered. What do you think are some of the motivations behind gambling (that is, how do people justify gambling)?

A home washing machine removes grease and dirt from clothes at a first-order removal rate in which 12% of the grease and dirt is removed per minute of washing. The washing machine holds 50 L of water and has a wash cycle of 5 minutes before discharging the wash water. What is the concentration of dirt and grease in mg/L in the discharge water if the clothes initially contain 0.5 g of dirt and grease? (Hint: removal rate is not k. You must solve for k!)

Your engineering consulting firm has been retained to develop an absorption process that removes 98% of the SO2 from an air stream. The column operates at 30 oC and 5 atm. The design assumes 250 mol/min of an air stream contaminated with 10 mol% SO2 fed to the bottom of the column. Regenerated absorbent is fed to the top of the column. The absorbent is an aqueous solution, where the absorbent species is non-volatile. Some water will vaporize based on its vapor pressure at 30 oC (approximately 35 mmHg), so the treated gas will contain air, residual SO2, and water vapor. The absorbent stream leaving the column contains SO2 in a molar ratio of 15:1 (solvent:SO2). The absorbent is recovered in a stripping column operating at 120 oC and 2 atm. The stripping gas is 250 mol/min steam which is fed to the bottom of the stripping column and removes 98% of the SO2. Some of the steam condenses in the absorbent phase at a ratio of 2:1 (aqueous solvent:condensate). The absorbent is non-volatile, so the exiting steam only contains water and SO2. Your task is to analyze this process. Follow the problem solving procedure outlined in your notes and the textbook. You must: prepare a fully labeled process flow diagram, conduct a degree of freedom analysis on each system (overall, absorber, and stripper), clearly identify unknown variables and units, write all appropriate equations (materials balances, etc.) needed to solve for the unknowns, solve the problem.