Chapter 1: Introduction to the Monty Hall Problem

Recently, I tuned into a podcast that was highly recommended on this platform. During one of the episodes, which revolved around big-O notation, the Monty Hall dilemma surfaced. The hosts touched on it briefly, confessing their confusion before moving on. This prompted me to delve into this topic, as I believe it’s crucial for engineers to fully comprehend such concepts.

Here’s my understanding of the problem. After some contemplation, I found that it isn't overly complex once you break it down.

If you’re not familiar with it, the Monty Hall problem presents a scenario with three doors. Behind one door lies a prize, while the other two conceal nothing. You select a door, hoping it hides the prize. Subsequently, a host—who knows what's behind each door—opens one of the remaining doors that does not have the prize. You are then faced with a choice: stick with your original selection or switch to the other unopened door. Which option should you choose? Does it even make a difference?

Many people initially think that, since one door is revealed to be empty, the odds between the two remaining doors are now equal at 50-50. However, this intuition is misleading. Statistically, it’s more beneficial to switch doors. To solidify our understanding, let’s begin with the hypothesis that there’s a 50% chance of winning, regardless of whether we switch or stay.

With our hypothesis in place, let’s conduct an experiment.

The first video titled Monty Hall Problem (best explanation) - Numberphile - YouTube provides an excellent overview of the Monty Hall problem and explores its intricacies.

In this experiment, we can gain additional insights by randomly alternating between staying with our initial choice and switching doors.

The second video, Monty Hall Problem - Numberphile - YouTube, further elaborates on the reasoning behind the best choice.

After running the experiment, it becomes evident that our initial hypothesis needs refinement. It appears that switching doors is the better strategy. Perhaps a new intuition has emerged: winning by staying with your original choice is only possible if you guessed correctly the first time.

Chapter 2: Analyzing the Problem

To analyze this further, let’s visualize a timeline of possibilities. With three doors and one prize, we can evaluate the probability of selecting the correct door on the first try.

Let’s pause here for a more thorough breakdown. There are two potential outcomes based on your initial choice:

1. Universe 1: You select the correct door, which occurs 33.3% of the time. In this case, the host can open either of the other two doors, leading to two scenarios:
   • Universe 1.1: You stay with your initial pick (50% chance of winning).
   • Universe 1.2: You switch (100% chance of losing).
2. Universe 2: You choose incorrectly, which happens 66.7% of the time. Here, the host can only open the door without the prize, leaving the other door as the only potential winner:
   • Universe 2.1: You stay with your wrong choice (100% chance of losing).
   • Universe 2.2: You switch (100% chance of winning).

Chapter 3: Mathematical Analysis

The crux of our inquiry is to determine whether switching or staying yields a higher probability of winning.

Let’s compute the chances for both options:

• Probability of winning by staying:

[

P(win | stay) = frac{P(win) times P(stay | win)}{P(stay)}

]

After evaluating, we find:

[

P(win | stay) = frac{1/2 times 1/3}{1/2} = 1/3

]

• Probability of winning by switching:

[

P(win | switch) = frac{P(win) times P(switch | win)}{P(switch)}

]

This leads to:

[

P(win | switch) = frac{1/2 times 2/3}{1/2} = 2/3

]

From our mathematical investigation, it’s clear that switching doors provides a two-thirds chance of winning, compared to just one-third if you choose to stay.

Chapter 4: Logical Reasoning

If you always choose to switch, you will win whenever your initial pick is incorrect, which occurs approximately 66.67% of the time. Conversely, if you never switch, your chances improve only if you initially select the right door, which is only a 33.3% probability.

Chapter 5: Expanding the Concept

To deepen our understanding, consider extending the problem to an extreme scenario: imagine there are one million doors with one prize. After selecting a door, the host would then open all but one remaining door. If you believe you made the right choice initially, the odds are heavily against you. In this case, switching would give you a staggering 999,999 out of 1,000,000 chances of winning.

This perspective aids in grasping the fundamental principle behind the Monty Hall problem, reinforcing the idea that switching maximizes your chances.

Now that our hypothesis aligns with the experimental data, we can conclude this exploration of the Monty Hall dilemma.

This essay is part of a series on mathematical topics published in Cantor's Paradise, a weekly Medium feature. Thank you for your attention!