Question $2$ of $5$

Problem $2$ FIFA Group Stage

In the group stage of the FIFA world cup each team plays three games with the other members of its group. A win earns $3$ points, a draw $($tie$)$ earns $1,$ and a loss earns $0$ points.

What are the different numbers of points a team can end up with at the end of group play $($after $3$ games$)?$ Draw an event tree to work this out.

If one team wins all of $3$ of its games, what are the possible outcomes for the other three teams? Draw an event tree to work this out.

If in every game it is equally likely that a team will win, draw, or lose, what is the probability that a team ends up with each of the possible number of points? Question $3$ of $5$

Problem $3$ Project Management

A company needs to visualize the task dependencies in a complex project. Each node in the DAG represents a task, and an arrow from node $1$ to node $2$ means that the task represented by the first node must be completed before the task represented by the second node can be started.

Here is a list of the tasks for one of the projects, along with the dependencies between them:

Task A: No dependencies

Task B: Dependent on Task A

Task C: Dependent on Task A

Task D: Dependent on Task B and Task C

Task E: Dependent on Task D

Your task is use Python to draw a DAG to represent these dependencies, and then analyze it to determine the order in which the tasks must be completed. Question $4$ of $5$

Problem $4$ Playing Cards

Suppose you have a deck of $52$ playing cards and you draw $2$ cards at random $($without replacement$).$

a$.$ The probability that you will draw two cards of the same rank $($e$.$g$.,$ two $7$s$,$ two Kings, etc.$)$ from a standard deck of $52$ cards is

$$

$13$

$\backslash$times

$4$

$$

$2$

${}_{52}C_{?}52C_{2}0\mathrm{.}0588$

Perform a data generation process and verify that the simulated probability is nearly close to this value for a sufficiently large simulation runs. Explain each code line and its output.

b$.$ Suppose you draw $5$ cards at random $($without replacement$).$ What is the probability that you will draw a full house OR four of a kind?

Reminder: A full house means $3$ of one value card and $2$ of another value card $($for example, three kings and two $8$s$)$; four of a kind means $4$ cards of the same value.

NOTE: even if you can't fully solve this, describe the approach you would take and what principles or theorems you might make use of$,$ and make as much progress as you can.