Take a deep breath before attempting this problem. In the book Innumeracy [22], John

Allen Paulos writes:

Now for better news of a kind of immortal persistence. First, take a deep

breath. Assume Shakespeare's account is accurate and Julius Caesar gasped

["Et tu, Brute!"] before breathing his last. What are the chances you just

inhaled a molecule which Caesar exhaled in his dying breath?

Assume that one breath of air contains 1022 molecules, and that there are 1044 molecules

in the atmosphere. (These are slightly simpler numbers than the estimates that Paulos

gives; for the purposes of this problem, assume that these are exact. Of course, in reality

there are many complications such as di?erent types of molecules in the atmosphere,

chemical reactions, variation in lung capacities, etc.)

Suppose that the molecules in the atmosphere now are the same as those in the atmosphere when Caesar was alive, and that in the 2000 years or so since Caesar, these

molecules have been scattered completely randomly through the atmosphere. You can

also assume that sampling-by-breathing is with replacement (sampling without replacement makes more sense but with replacement is easier to work with, and is a very good

approximation since the number of molecules in the atmosphere is so much larger than

the number of molecules in one breath).

Find the probability that at least one molecule in the breath you just took was shared

with Caesar's last breath, and give a simple approximation in terms of e.

56. A widget inspector inspects 12 widgets and finds that exactly 3 are defective. Unfortunately, the widgets then get all mixed up and the inspector has to find the 3 defective

widgets again by testing widgets one by one.

(a) Find the probability that the inspector will now have to test at least 9 widgets.

(b) Find the probability that the inspector will now have to test at least 10 widgets.

For each part, decide whether the blank should be filled in with =, , and give a

clear explanation. In (a) and (b), order doesn't matter.

(a) (number of ways to choose 5 people out of 10) (number of ways to choose 6

people out of 10)

(b) (number of ways to break 10 people into 2 teams of 5) (number of ways to

break 10 people into a team of 6 and a team of 4)

(c) (probability that all 3 people in a group of 3 were born on January 1) (probability that in a group of 3 people, 1 was born on each of January 1, 2, and 3)

Martin and Gale play an exciting game of "toss the coin," where they toss a fair coin

until the pattern HH occurs (two consecutive Heads) or the pattern TH occurs

4. A Markov chain with 4 states (1, 2, 3,4) has the onestep transition matrix 12 ---00 1330 P=3131 21? 0055 a. Write down the equilibrium equations. [4 marks] b. Solve these equations. [7 marks] c. If the Markov chain has been running a long time, what is the proportion of time the Markov chain spends in State 3? [1 mark] A Markov chain has the states 0, 1, 2, 3, 4, 5, 6. At time 0 the chain is in state 0. Then, for any given time n 2 1, a fairsix sided die is tossed and the state of the chain at time n. is the largest number which has been obtained in an. die tosses. (a) Why is this system a Markov chain? (b) Determine the transitiOn probability matrix of the Markov chain. (c) Determine the classes of the Markov chain. Are any of the classes closed in the sense that once that class is entered the chain can never exit that class? (d) Suppose now that that we add the rule to the system that if we get a six in two tosses in a row, then the system goes back to state 0. Is the system then still a Markov chain with the states 0, 1, 2, 3, 4, 5, 6? Motivate your answer. 1. (a) Define the Markov property of a discrete Markov chain. [5 marks] (b) Explain the difference between a homogeneous and inhomoge- neous Markov chain. [5 marks] (c) A three-state Markov chain has transition probability matrix, P, given below 1 2 3 1/ 0.8 0.1 0.1\\ P -2 0.15 0.75 0.1 3 0.06 0.04 0.9 If for all large n /0.3 0.2 0.5 0.3 0.2 0.5 0.3 0.2 0.5 what is the long-term proportion of time the chain spends in each of the three states? [5 marks) (d) State, with proof, whether the Markov chain in the previous ques- tion is reversible. [5 marks] (e) Assume that rows 1, 2 and 3 in the transition probability matrix, P correspond to transitions from states r, s and q, respectively of the chain. If the chain is in state q at some point i, what is the probability it was in state r at point i - 2? [5 marks]al d) Of the 1682 respondents, 493 identified themselves as Catholic. Statistic e) Four hundred thirty of the respondents reported that they had never married. Statistic (Noticing that a part of the calculation here corresponds to the calculation already done in d can save you a bit of work) f) The proportion of respondents who voted for the Conservative Party of Canada in the prior fed election was 0.36. Statistic (Noticing that a part of the calculation here corresponds to the calculation already dane in d can save you a but of work.)