代做MAST20004 Assignment 1, S1 2024代写留学生Matlab程序

日期：2025-02-20 02:18

MAST20004 Assignment 1, S1 2024

Question 1

Old McDonald had a farm. On that farm he had 6 pigs, 5 cows, 7 chickens, 5 sheep, and 1 dog. An experiment involves selecting (“uniformly at random”) one animal from the farm and observing what kind of animal it is.

(a) The sample space Ω for this experiment has 5 elements. List them.

(b) Give the individual probabilities for each of the sample points in Ω.

(c) Find the probability of the event, M, that the selected animal is a mammal.

(d) Evaluate the probability of selecting a pig or cow, given that we select a mammal.

(e) Suppose instead that the experiment involves selecting two animals (uniformly at ran-dom) from the farm and moving them to a new farm. Find the probability that neither of the two animals is a mammal.

Question 2

In a game show called “Squad Game”, there are n contestants labelled 1 to n, in a circle.

Moving clockwise around the circle, starting with contestant 1, each contestant k who has not yet been eliminated nominates a contestant (which may be themselves) for possible elimination. Contestant k then rolls a fair 6-sided die, and if the result is a 6, then the nominated player is eliminated from the game. Play continues in this way until a fixed number < n of the players have been eliminated.

(a) Suppose that we watch this game and observe the labels of the players eliminated in the order in which they were eliminated. Describe a suitable sample space for this experiment.

(b) Suppose instead that we watch such a game and observe only the labels of the players eliminated. Describe a suitable sample space for this experiment.

(c) Suppose instead that we watch such a game and observe only whether player 1 is elimi-nated. Describe a suitable sample space for this experiment.

(d) Suppose that each player nominates a uniformly chosen player among those who have not yet been eliminated (i.e. if j < players have been eliminated then the player whose turn it currently is rolls a fair (n − j)-sided die to determine who they will nominate). Find the probability of each sample point in the three experiments (a)-(c) above.

(e) Find the probability that none of the first j nominations result in an elimination.

Question 3

Recall the game from question 2. Suppose that = 1, and that each contestant nominates the adjacent contestant who is anticlockwise of themselves.

(a) Find the probability that contestant 1 gets a second turn.

(b) Find the probability that contestant 1 is eliminated in the first n nominations.

(c) Find the probability that contestant 1 is the eliminated player.

(d) Which contestant is most likely to survive in this game, and why?

Question 4

Recall the game from question 2. Suppose that in each round (independent of the past) con-testant 1 nominates a uniformly chosen player other than themselves, while all other players always nominate contestant 1.

(a) If = 1, find the probability that contestant 1 is eliminated.

(b) Now suppose that n > = 3. Find the probability that player 1 is eliminated.