You and a friend are playing a game. You alternate turns rolling a single die, and the first person to roll a 1 or a 2 wins. Your friend goes first.

a. What’s the probability that the game ends in three rolls or fewer?

b. What’s the expected number of rolls?

c. What’s the probability that your friend wins?

Total marks: 50, plus 10 bonus marks for an answer in PLAN/DO/REPORT format.

An Elevating Business. The annual global market for selling elevators is worth $40bn and the annual global market for maintaining them is also worth $40bn. Just 5 companies have 80% of the market for sales: Kone, Otis, Schindler, Thyssenkrupp and Hitachi, however they have only 40% of the market for maintenance since a large number of small companies maintain elevators even though they don’t sell them. Thyssenkrupp is one of the most innovative elevator suppliers, having recently developed technology that allows elevators to move sideways and well as up and down. Another major innovator is Kone that has recently developed extra strength cables which allow very long travel heights, suited to new buildings over 100 stories high in Asia and the Middle East.

Thyssenkrupp is considering selling its elevator division, to cover financial problems in other divisions of the company. You are a financial analyst, assessing the probability of Thyssenkrupp’s elevator division being sold and to whom and also assessing the revenues if it is sold. Based on your experience in the elevator business, you estimate the following probabilities.

The probability of global sales increasing by > 2% next year is 0.25; and the probability of global maintenance revenues increasing by > 2% next year is 0.2. If at least one of these markets increases by > 2% next year, Thyssenkrupp will not sell its elevator division.

If Thyssenkrupp does want to sell, it could plan on selling to: 

a private equity company with a probability of 0.35. This would bring a quick infusion of cash to Thyssenkrupp    to assist its other divisions, 

Kone, with a probability of 0.45, resulting in a very large and innovative company, 

Otis, Schindler or Hitachi with a probability of 0.2.

If Thysennkrupp wants to sell to Kone, the deal may be prevented by European regulators with a probability of 0.65, since it would result in a single very large European elevator company which could constitute a monopoly.
(a) (20) Draw a probability tree to represent the above situation.
(b) (5) Which method of probability assessment was used to estimate the above probabilities?
(c) (10) What is the probability Thyssenkrupp does actually sell its elevator division to Kone?
(d) (15) You estimate that if Kone buys Thyssenkrupp’s elevator division the revenues will be as follows:

What is your estimate of the mean and standard deviation of the total revenues of the combined company assuming that Kone buys Thyssenkrupp’s elevator division?

1. The maximum amount of time a guest at the Holiday Inn can wait for an elevator is 4 minutes. Assuming the wait time follows a uniform distribution:

1. How long should a guest expect to wait?

2. What is the standard deviation of wait times?

3. What is the probability that a guest waits exactly 2 minutes for the elevator?

4. What is the probability that a guest waits more than 3 minutes?

5. What is the probability that a guest waits less than 90 seconds?

6. What is the probability that a guest waits between 1 minute and 2:30 for the elevator?

In a volleyball game between two teams , A and B, the game will be over if a team wins two out of three sets. In any set, team A has a 60 percent chance of winning the sets.

1. Model this game as a Markov chain. Hint: Use (WA, WB) as the states where WA is the number of sets A wins and WB is the number of sets B wins. For example state (1,0) means A won 1 set and B won 0 set. Of course, the game starts with a state of (0,0).
2. Identify each state of this markov chain.
3. Compute the expected number of sets before team A can win the game.
4. Compute the expected number of sets before team B can win the game.
5. Compute the probability that team A can win the game.
6. Compute the probability that team B can win the game.

A group of six friends play some games of ping-pong with these results: Amy beats Bob Bob beats Carl Frank beats Bob Amy beats Elise Carl beats Dave Elise beats Carl Elise beats Dave Frank beats Elise Frank beats Amy Consider the relation R = {hx, yi : x has beaten y}. (a) Draw the directed graph G representing R. (b) Is R reflexive? Irreflexive? Symmetric? Asymmetric? Antisymmetric? Transitive? An equivalence? An order? (c) The players want to rank themselves. Find every possible topological order of G. (d) In order to have a definitive ranking, the players want there to be only one possible topological order. Which two players should face each other? (e) The transitive closure of R (R+), is an order. Is it partial or total?

An elevator has a placard stating that the maximum capacity is 2310 lblong dash15 passengers.​ So, 15 adult male passengers can have a mean weight of up to 2310 divided by 15 equals 154 pounds. If the elevator is loaded with 15 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 154 lb.​ (Assume that weights of males are normally distributed with a mean of 161 lb and a standard deviation of 35 lb​.) Does this elevator appear to be​ safe? The probability the elevator is overloaded is?

Sally plays a game and wins with probability p. Every week, she plays until she wins two games, and then stops for the week. Sally calls it a "lucky week" if she manages to achieve her goal in 7 or less games.

a) If p = 0.2, what's the probability that Sally will have a "lucky week" next week?

b) What's the probability of exactly 3 "lucky weeks" in the next 5 weeks? What's the expected number of "lucky weeks" Sally will have in the next 10 weeks?

c) Let X be the number of games Sally plays in a week, and let q = 1 – p. Find the expectation E[X].

d) If Sally pays $1 to play each game, and gets $5 for each game she wins, what's her expected earning at the end of each week?

Sally plays a game and wins with probability p. Every week, she plays until she wins two games, and then stops for the week. Sally calls it a "lucky week" if she manages to achieve her goal in 7 or less games.

a) If p = 0.2, what's the probability that Sally will have a "lucky week" next week?

b) What's the probability of exactly 3 "lucky weeks" in the next 5 weeks? What's the expected number of "lucky weeks" Sally will have in the next 10 weeks?

c) Let X be the number of games Sally plays in a week, and let q = 1 – p. Find the expectation E[X].

d) If Sally pays $1 to play each game, and gets $5 for each game she wins, what's her expected earning at the end of each week?

An elevator has a placard stating that the maximum capacity is 1328 lblong dash8 passengers.​ So, 8 adult male passengers can have a mean weight of up to 1328 divided by 8 equals 166 pounds. If the elevator is loaded with 8 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 166 lb.​ (Assume that weights of males are normally distributed with a mean of 170 lb and a standard deviation of 28 lb​.) Does this elevator appear to be​ safe?

The probability the elevator is overloaded is?

(Round to four decimal places as​ needed.

An elevator has a placard stating that the maximum capacity is 1560 lb—10 passengers.​ So, 10 adult male passengers can have a mean weight of up to 1560/10=156 pounds. If the elevator is loaded with

10 adult male​ passengers, find the probability that it is overloaded because they have a mean weight greater than 156 lb.​ (Assume that weights of males are normally distributed with a mean of 165 lb and a standard deviation of 28 lb​.) Does this elevator appear to be​ safe?

The probability the elevator is overloaded is ___​(Round to four decimal places as​ needed.)

Does this elevator appear to be​ safe?

A. Yes, 10 randomly selected people will always be under the weight limit.

B. ​No, there is a good chance that 10 randomly selected people will exceed the elevator capacity.

C. ​No, 10 randomly selected people will never be under the weight limit.

D. ​Yes, there is a good chance that 10 randomly selected people will not exceed the elevator capacity.