1.Exercise 4) A fair coin is tossed. If it lands heads, a fair four-sided die
is thrown (with values 2,3,4,7). If
2,3,4,7). If it lands tails, a fair six-sided die is
thrown (with values 3,4,5,6,7,9). Regardless of which die is used, Alice
eats n grains of rice, where n is the largest prime factor of the die result
(for example, the largest prime factor of 9 is 3).
(a) What is the conditional probability that the coin lands heads, given
that Alice eats three grains of rice?
(b) Suppose that the entire experiment is conducted twice on the following day (starting with a new coin toss on the second run-through).
What is the conditional probability that the coin lands heads on both
run-throughs, given that Alice eats a total of five grains of rice during the two run-throughs?
(Do not count the two grains from part (a) in part (b); we assume
two brand new experiments, each with a new coin toss. Start your
solution by defining a suitable partition of the sample space. Please
use an appropriate notation and/or justification in words, for each
value that you give as part of your solution.)
Exercise 5) Alice and Bob throw an unfair coin repeatedly, with probability 2/5 of landing heads. Alice starts with £2 and Bob starts with £3 .
Each time the unfair coin lands heads, Alice gives Bob £1 . Each time
the unfair coin lands tails, Bob gives Alice £1 . The game ends when one
player has £5 .
(a) Draw a labelled Markov chain describing the problem, and write
down a transition matrix P. Write down the communication classes,
and classify them as either recurrent or transient.
(b) Using the transition matrix, calculate the probability that Alice loses
all of her money in exactly four tosses of the unfair coin.
(c) Calculate the (total) probability that Alice loses all of her money
(before Bob loses all of his).
(d) Calculate the expected (mean) number of tosses of the unfair coin,
for the game to end.
2.Describe the parallels between cell differentiation, nutrient supply, and final cellular state for the majority of cells involved in the
he majority of cells involved in the epidermis, hair follicles/hair growth, and nail synthesis/nail growth.
How does the structure of spongy bone support the function of the red bone marrow contained within said bone?
How do bones, synovial fluid, tendons, ligaments, and articular cartilage work together to allow synovial joint movement? Please explicitly include the bone feature that allows for tendon/ligament attachment.
What is the difference in function between the vertebrae and the intervertebral disks in the vertebral column? Please discuss both the connective tissue subtype for each of these features and how the composition of their extracellular matrix contributes to function
Describe the extracellular matrix of bone to someone who isn't familiar with anatomy (this should be a general description of appearance/tactile nature, no specific mention of cells or function is needed)
How does hair contribute to the sensory role of the integumentary system? (A short explanation is all that is needed, no in-depth description of associated structures is required for full credit)
Back when he was 16, Jordan left eye orbit floor was fractured in a bizarre fencing accident, leaving a few fragments behind inside his skull. Assuming the floor was punctured from above in the maxillary bone and no additional bones were broken, where did the bone fragments end up in his skull? (This should be a particular space in the skull, not a separate bone where the fragments are embedded
The skeletal and integumentary systems rely heavily on collagen for function - describe one or more features other than ligaments that involve collagen, and how you would expect function to change in someone with EDS that affected the relevant subtype of collagen
3.Question 1: What is a player’s « reaction function » in a Bertrand game ?
Question 2: What is a subgame
subgame perfect Nash equilibrium?
Question 3: In which situations should we need the mixed extension of a game?
Question 4: Find, if any, all Nash equilibria of the following famous matrix game:
U (2,0) (3,3)
D (3,4) (1,2)
Question 5: What is the difference between a separating equilibrium and a pooling equilibrium
in Bayesian games?
Question 6: Give another name for, if it exists, the intersection of the players’ best-response
« functions » in a game?
Question 7: assuming we only deal with pure strategies, the Prisoner’s Dilemma is a situation
No Nash equilibrium One sub-optimal Nash equilibrium
One sub-optimal dominant profile No dominant profile
Question 8: If it exists, a pure Nash equilibrium is always a profile of dominant strategies:
Question 9: All games have at least one pure strategy Nash equilibrium:
Question 10: If a tree game has a backward induction equilibrium then it must also be a Nash
equilibrium of all of its subgames:
Question 11: The mixed Nash equilibrium payoffs are always strictly smaller than the pure
Nash equilibrium payoffs:
Question 12: Which of the following statements about dominant/dominated strategies is/are
I. A dominant strategy dominates a dominated strategy in 2x2 games.
II. A dominated strategy must be dominated by a dominant strategy in all games.
III. A profile of dominant strategies must be a pure strategy Nash equilibrium.
IV. A dominated strategy must be dominated by a dominant strategy in 2x2 games.
I, II and IV only I, II and III only II and III only
I and IV only I, III and IV only I and II only
Question 13: A pure strategy Nash equilibrium is a special case of a mixed strategy Nash
Question 14: Consider the following 2x2 matrix game:
U (3,2) (2,4)
D (-1,4) (4,3)
The number of pure and mixed Nash equilibria in the above game is:
Exercise (corresponding to questions 15 to 20 below): assume a medical doctor (M)
prescribes either drug A or drug B to a patient (P), who complies (C) or not (NC) with each of
this treatment. In case of compliance, controlled by an authority in charge of health services
quality, the physician is rewarded at a level of 1 for drug A and 2 for drug B. In case of noncompliance, the physician is « punished » at -1 level for non-compliance of the patient with
drug A and at -2 level for non-compliance with drug B. As for the compliant patient, drug A
should give him back 2 years of life saved and drug B, only 1 year of life saved. When noncompliant with drug A, the same patient wins 3 years of life (due to avoiding unexpected
allergic shock for instance), and when non-compliant with drug B, the patient loses 3 years of
Question 15: You will draw the corresponding matrix of the simultaneous doctor-patient game.
Question 16: Find, if any, the profile(s) of dominant strategies of this game.
Question 17: Find, if any, the pure strategy Nash equilibrium/equilibria of this game.
Question 18: Find, if any, the mixed strategy Nash equilibrium/equilibria of this game.
Questions 19 and 20: Now the doctor prescribes first, then the patient complies or not: draw
the corresponding extensive-form game (= question 19) AND find the subgame perfect Nash