The Monty Hall Problem

Posted on Thu 01 December 2022 in simulation

The Monty Hall Problem and Streamlit

Once I learned about Streamlit.io, I couldn't resist the opportunity to write a simulation of the Monty Hall problem!

The Monty Hall problem involves a game show and prizes! On the game show, contestants were given the choice of three doors. One door conceals a fabulous prize!

Contests are asked to select a door.

Then comes the twist -

The host opens one of the two remaining doors and asks if the contestant would like to switch to the remaining door.

The question that emerges is: Switch or Stay?

What should the contestant do?

The contestent should go with the switch strategy - this will result in a win 66% of the time. In fact, the only way the contestant looses with the switch strategy is if they originally selected the prize door.

With the Monty Hall problem, we can find the answer in three different ways!

  1. Using Bayes Theorem and probability theory
  2. Hosting a game show and keeping track of the winners
  3. Constructing a Monte Carlo simulation

Here's my streamlit simulation.

In this simulation, we construct a data frame to keep track of our game. For each trial, we need to perform four steps:

  1. Selecting a door for the prize
  2. Selecting a door for the contestant's choice
  3. Determine if the Stay strategy won (or not).
  4. Determine if the Switch strategy won (or not).

In addition, we want to tabulate the percentage of wins go to each strategy over time.

Steps 1 and 2:

We simulate the door choices using randomint.

 select_door = random.randint(1,3)

Steps 3 and 4:

Write a short function to determine if the strategy won. Wins are represented by 1's and losses by 0's. Using this, we can easily tabulate the wins in graph.

def switch_strat(init_door: int, win_door: int)->int:
    """Switch strategy 
    Args:
        init_door (int): initial door selected
        win_door (int): prize door
    Returns:
        int: 0 or 1 to indicate win-loose
        """
    if init_door != win_door:
        return 1
    else: 
        return 0

Displaying the first 5 trials

The first five trials and their outcomes are displayed in a data frame. Simply looking at the first few trials can be deceptive, so the cumulative percentage wins are displayed in a graph.

We can add additional trials and see how the win percentages change as the number of trials increases. (I had to set some session variables so that the data refreshes correctly.)

A visual display

Finally, I installed a graph to display how the percentage of wins change over time for each strategy.

st.line_chart(results)

We can use this chart to examine the long term behavior and see that (eventually) the win percentage for the Switch strategy converges to 66%.

Even more excitingly, we could easily expand this simulation to modified versions of the Monty Hall problem or the Birthday Problem!

streamlit