Three people get into an empty elevator at the first floor of a building that has 10

floors. Each presses the button for their desired floor (unless one of the others has

already pressed that button). Assume that they are equally likely to want to go to

floors 2 through 10 (independently of each other). What is the probability that the

buttons for 3 consecutive floors are pressed?

How many ways are there to permute the letters in the word MISSISSIPPI?

2. (a) How many 7-digit phone numbers are possible, assuming that the first digit can't

be a 0 or a 1?

(b) Re-solve (a), except now assume also that the phone number is not allowed to start

with 911 (since this is reserved for emergency use, and it would not be desirable for the

system to wait to see whether more digits were going to be dialed after someone has

dialed 911).

3. Fred is planning to go out to dinner each night of a certain week, Monday through

Friday, with each dinner being at one of his ten favorite restaurants.

(a) How many possibilities are there for Fred's schedule of dinners for that Monday

through Friday, if Fred is not willing to eat at the same restaurant more than once?

(b) How many possibilities are there for Fred's schedule of dinners for that Monday

through Friday, if Fred is willing to eat at the same restaurant more than once, but is

not willing to eat at the same place twice in a row (or more)?

4. A round-robin tournament is being held with n tennis players; this means that every

player will play against every other player exactly once.

(a) How many possible outcomes are there for the tournament (the outcome lists out

who won and who lost for each game)?

(b) How many games are played in total?

5. A knock-out tournament is being held with 2n tennis players. This means that for each

round, the winners move on to the next round and the losers are eliminated, until only

one person remains. For example, if initially there are 24 = 16 players, then there are

8 games in the first round, then the 8 winners move on to round 2, then the 4 winners

move on to round 3, then the 2 winners move on to round 4, the winner of which is

declared the winner of the tournament. (There are various systems for determining who

plays whom within a round, but these do not matter for this problem.)

(a) How many rounds are there?

(b) Count how many games in total are played, by adding up the numbers of games

played in each round.

(c) Count how many games in total are played, this time by directly thinking about it

without doing almost any calculation.

Hint: How many players need to be eliminated?

6. There are 20 people at a chess club on a certain day. They each find opponents and

start playing. How many possibilities are there for how they are matched up, assuming

that in each game it does matter who has the white pieces (in a chess game, one player

has the white pieces and the other player has the black pieces

Q 4. (28 points total 4 points each/ Imagine a survey completed by a random sample of 1682 Albertans provided the information in each of the statements below. For each statements identify the relevant sample statistic and construct a 95% confidence interval for the corresponding population parameter using the information provided. The first confidence interval requires a careful interpretation. The remaining confidence intervals do not require a reported interpretation.) Round your final answers to four places behind the decimal. You may report your confidence intervals using , or as two numbers. al (4 points) The average occupational prestige score was 49.16 with standard deviation 13.52. Statistic az (4 points) Interpret the confidence interval you calculated in part an. b) The respondents reported watching on average 2.86 hours of electronic-media per day, with a standard deviation of 2.20. Statistic c) The average number of children was 1.81 with a standard deviation of 1.67. Statistic(9) Stokes' Theorem: Stokes' Theorem converts a circulation calculation into a surface calculation through the formula where F is a given vector field, Curl F = V x F. S is a surface enclosed by the closed curve C, and 7 is the outward pointing unit normal. (a) Find the circulation of the vector field F = yi - r } on the circle Cox' + y' = 9, = =0 directly. (b) Find the circulation of the vector field in part (a), by converting the calculation to a surface calculation using Stoke's theorem, on the surface S :x+y' + 2 =9,2 20.Problem 6. Below is the transition probability matrix of a Markov chain (X.) with six states. Determine the communication classes of this Markov chain, and determine for each class whether it is recurrent or transient. Is this Markov chain ergodic? Why or why not? 0.9 O 0.2 0.2 04 0 0 0 05 0 0.5 0