Suppose that there are two hats in front of you.

• Hat 1 contains 4 green balls and 6 orange balls.

• Hat 2 contains 3 green balls, 6 orange balls, and a purple ball.

• (a) Suppose you draw one ball from each hat. The outcome of interest is the colour of each of the drawn balls. How many elements are in the sample space of this experiment?

• (b) Write out the complete sample space for the experiment above.

• (c) What is the probability that you draw two balls of the same colour?

(d) Consider the following variables.

• Suppose you draw from Hat 1, without replacement, until you get a green ball. Let X be the number of draws until you get a green ball.

• Suppose you draw a ball from Hat 1 and a ball from Hat 2. Let Y be the number of drawn orange balls.

• Suppose you draw a ball from Hat 1 and a ball from Hat 2. Let Z be the number of drawn green balls.

• Suppose you draw 6 balls from Hat 2, with replacement. Let W be the number of times a green ball is drawn.

  Determine which of the above variables follow a binomial distribution. Also, for each binomially-distributed variable, determine the parameters n and p.

1. Keno: Keno game is a game with 80 numbers 1, 2, … , 80 where 20 numbered balls out of these 80 numbers will be picked randomly. You can pick 4, 5, 6, or 12 numbers as shown in the attached Keno payoff / odds card.

When you pick 4 numbers, there is this 4-spot special that you place $2.00 in the bet, and you are paid $410 if all your 4 numbers are among the 20 numbers, or your are paid $4.00 if 3 of the 4 numbers are among the 20 numbers chosen from the 80 numbers.

The number of ways of picking 20 numbers from 80 is C(80, 20) = 80! / (60! 20!).

The number of ways that all your 4 numbers are among the 20 numbers is: C(76, 16) (why?) = 76! / (60! 16!).

The probability that your 4 numbers bingo is C(76, 16) / C(80, 20), and the theoretical payoff should be C(80, 20) / C(76, 16) = 80! * 16! / (76! * 20!) = (80 * 79 * 78 * 77 ) / (20 * 19 * 18 * 17 ) = $326.4355…

1. (6%) Based on this computation, is the payoff fair? Explain!
2. (6%) The payoff for 3 numbers in your 4 chosen numbers appear in the 20 numbers is $4.00. Is that a fair payoff (how is the number of ways of 3 numbers matching related to the number of ways of 4 numbers matching?)?

10. (12%) Keno: Keno game is a game with 80 numbers 1, 2, … , 80 where 20 numbered balls out of these 80 numbers will be picked randomly. You can pick 4, 5, 6, or 12 numbers as shown in the attached Keno payoff / odds card. When you pick 4 numbers, there is this 4-spot special that you place $2.00 in the bet, and you are paid $410 if all your 4 numbers are among the 20 numbers, or your are paid $4.00 if 3 of the 4 numbers are among the 20 numbers chosen from the 80 numbers. The number of ways of picking 20 numbers from 80 is C(80, 20) = 80! / (60! 20!). The number of ways that all your 4 numbers are among the 20 numbers is: C(76, 16) (why?) = 76! / (60! 16!). The probability that your 4 numbers bingo is C(76, 16) / C(80, 20), and the theoretical payoff should be C(80, 20) / C(76, 16) = 80! * 16! / (76! * 20!) = (80 * 79 * 78 * 77 ) / (20 * 19 * 18 * 17 ) = $326.4355… (a) (6%) Based on this computation, is the payoff fair? Explain! (b) (6%) The payoff for 3 numbers in your 4 chosen numbers appear in the 20 numbers is $4.00. Is that a fair payoff (how is the number of ways of 3 numbers matching related to the number of ways of 4 numbers matching?)?

1. The song-length of tunes in the Big Hair playlist of a certain Statistics professors mp3-player vary from song to song. This variation can be modeled by the Normal distribution, with a mean song-length of μ=4.1 minutes and a standard deviation of σ=0.7 minutes. Note that a song that has a length of 4.5 minutes is a song that lasts for 4 minutes and 30 seconds.

From the time he set his mp3-player to shuffle, there has been 16 songs randomly chosen and played in succession. What is the chance that the 16-th song played is the 8-th to be longer than 4.1 minutes? Enter your answer to four decimals.

2. The number of people arriving per hour at the emergency room (ER) of a local hospital seeking medical attention can be modeled by the Poisson distribution, with a mean of 10 people per hour.

The inter-arrival time, X X, is defined as the time that passes between successive arrivals of patients seeking medical attention. It has been 5 minutes since the last person seeking medical attention arrived at the ER. What is the probability that at least 11 minutes (in total) will pass until the next medical-attention-seeking person passes through the ER doors? Use four decimals in your answer.

3.The number of customers entering a 24-hour convenience store every 10-minutes can be modeled by the Poisson distribution with a a mean of λ=5.2 customers. You are to look at the amount of time passing between successive customers entering the convenience store, represented by X

Compute the probability at most 3 minutes will pass between the arrival of one customer and the next customer. In answering this question, be sure to use your answer in (a) to two decimals. Enter your answer using four decimals

At least 1 minute has passed since the last customer entered the store. What is the probability that in total, at least 4 minutes will pass until the next customer enters this store? Use four decimals in your answer.

STRAIGHT FROM THE BOOK

Roulette In the casino game of roulette there is a wheel with 19 black slots, 19 red slots, and 2 green slots. In the game, a ball is rolled around a spinning wheel and it lands in one of the slots. It is assumed that each slot has the same probability of getting the ball. This results in the table of probabilities below.

Fair Table Probabilities

You watch the game at a particular table for 130 rounds and count the number of black, red, and green results. Your observations are summarized in the table below.

Outcomes (n = 130)

The Test: Test the claim that this roulette table is not fair. That is, test the claim that the distribution of colors for all spins of this wheel does not fit the expected distribution from a fair table. Test this claim at the 0.01 significance level.

(a) What is the null hypothesis for this test?

H₀: The probabilities are not all equal to 1/3.H₀: p₁ = 19/40, p₂ = 19/40, and p₃ = 2/40.    H₀: p₁ = p₂ = p₃ = 1/3H₀: The probabilities associated with this table do not fit those associated with a fair table.

(b) The table below is used to calculate the test statistic. Complete the missing cells.
Round your answers to the same number of decimal places as other entries for that column.

(c) What is the value for the degrees of freedom?
(d) What is the critical value of χ²? Use the answer found in the χ²-table or round to 3 decimal places.
t_α =

(e) What is the conclusion regarding the null hypothesis?

reject H₀

fail to reject H₀    

(f) Choose the appropriate concluding statement.

We have proven that this table is fair.

The results of this sample suggest the table is not fair.    

There is not enough data to conclude that this table is not fair.

4. Given a population composed of dogs (40%) and cats (60%), in a sample of size of 10

Find the probability of 3 cats.

Find the probability of fewer than 3 dogs.

Find the probability of between 4 and 7 cats.

Find the probability of 2 cats and 2 dogs.

The settlement of a structure has the normally distributed probability density function with a mean of 26mm and a coefficient of a variation of 20%.

1) What is the probability that the settlement is less than 22mm?

2) What is the probability that the settlement is between 24mm and 29mm?

3) What is the probability that the settlement exceeds 31mm?

Please Provide R code as well

Use R to find probability (p-value). Find probability P(X>12.3), where X follows F-distribution with degree of freedom in numerator 4 and degree of freedom in numerator 10.

You are the controller for Bizbee Corporation, and a few days ago, you provided a draft of this year's financial statements to the chief executive officer (CEO) of the company, Mr. Bizbee. You rode up in the elevator with him today, and he began to quiz you about how you reported the company's investments in debt and equity securities. He said to you, "When I took accounting in college, investment securities were reported at historical cost. I remember what we paid for some of our investments, and the numbers on the financial statements don't match those amounts! What's going on? Be in my office this afternoon to explain!"

To get ready for your meeting, assemble the following:

• The valuation approach was used on the balance sheet for the investments. Provide an analysis on why you used this approach. Assume your company has only debt and equity securities where the equity interest is less than 20%. The company's debt securities are all classified as held to maturity, but it has both trading and available-for-sale equity securities.
• Assemble and evaluate which generally accepted accounting principles (GAAP) for investment securities changed from historical cost to the current valuation approach.
• Assess how investment securities should be valued on the company's financial statements.