Question $3$ of $5$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

$\frac{13{?}^4{C}_2}{{?}^{52}{C}_2}$~~$0\mathrm{.}0588$

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.

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.