Lecture 4 Finger Exercise

Question 1. Personnel at a certain company are classified as Employees, Managers, or Vice Presidents. At the end of each workday, an Employee has a prob_em chance of being promoted to Manager. Similarly, a Manager has a prob_mv chance of being promoted to Vice President after any given day, but also a prob_me chance of demotion back to Employee, where prob_mv + prob_me <= 1.

Implement in the function below a Monte Carlo simulation that estimates how many days on average it takes a new Employee to become a Vice President. Include your entire function definition: header and body. You may assume the random module is imported for you.

def estimate_timeline(prob_em, prob_mv, prob_me, num_trials):
    """
    Run a Monte Carlo simulation of num_trials trials to estimate how
    many days on average it takes a new Employee to become a Vice
    President. The parameters prob_em, prob_mv, and prob_me represent
    the promotion and demotion probabilities described above.
    """

For example:

• estimate_timeline(1, 1, 0, num_trials=100) should evaluate exactly to 2.
• estimate_timeline(1, 0.5, 0, num_trials=1000) should, with very high probability, evaluate to approximately 3.
• estimate_timeline(1, 0.5, 0.5, num_trials=1000) should, with very high probability, evaluate to approximately 4.
• estimate_timeline(0.1, 0.1, 0.1, num_trials=1000) should, with very high probability, evaluate to approximately 30.

Consider a band of thickness 1 centered around the circle's edge, i.e., the region contained between concentric circles with radii r - 0.5 and r + 0.5. The area of this band has the same numerical value as the circumference. Throw "darts" within a square that encompasses the outer-most circle, and use the hit rate of landing inside the band to estimate the circumference.

Implement this strategy in a function estimate_circumference(radius, num_darts), whose docstring is given below. You may assume the random module is imported for you.

def estimate_circumference(radius, num_darts):
    """
    Run a "dart-throwing" Monte Carlo simulation to estimate the
    circumference of a circle given its radius.

    Parameters:
        radius (float): The circle's radius.
        num_darts (int): The number of "darts" to throw within a square
            that encompasses such a circle.

    Use the hit rate in the scheme described above to estimate the
    circle's circumference, returning it as a float.
    """

For example:

• estimate_circumference(10, 10_000) should, with very high probability, evaluate to within 62.8 +/- 5.
• estimate_circumference(10, 1_000_000) should, with very high probability, evaluate to within 62.8 +/- 0.5.

Note: You should not hard-code any formulas involving a circle's circumference or area. However, you may wish to calculate distance between two points or from the origin. It is up to you what size square to use.