Why Switching is the Only Logical Choice
The Monty Hall problem is a brain-teaser based on the American game show Let’s Make a Deal. Imagine you are presented with three doors: behind one is a car, and behind the others are goats. You pick a door, say No. 1. The host, Monty Hall, who knows what is behind the doors, opens another door, say No. 3, which has a goat. He then asks you: “Do you want to switch to door No. 2?” Most people believe the odds are now 50/50, but switching actually doubles your chances of winning from 33.3% to 66.6%.
Visual Interpretation in Manim
The Manim animation breaks down the three-door scenario into a flow of Conditional Probability. It visualizes how the host’s knowledge changes the “weight” of the remaining unopened door.
- The Initial Pick: 1/3 vs 2/3 When you pick one door, you isolate 33% of the probability. The other 66% is tied to the two doors you didn’t pick.
- The Host’s Reveal: Filtering Information Monty Hall isn’t opening a door at random; he is intentionally removing a “losing” option from the 2/3 group.
- The Switch Advantage: Probability Transfer The animation shows the entire 2/3 (66%) probability shifting onto the single remaining unopened door. Switching captures the probability of both original unpicked doors.
Why it matters:
Our brains treat the new choice as a fresh 1-of-2 scenario, ignoring the history of the game. Manim shows that the history of the host’s choice is everything.
The Math Logic:
P(Win by Switching) = 1 – P(Initial Pick). Since P(Initial Pick) is 1/3, switching wins 2/3 of the time.
Note: This problem is a famous application of Bayes’ Theorem and serves as a classic example of Cognitive Bias. While the Monty Hall Problem seems simple, it famously fooled many PhD mathematicians when it was first published. It highlights the difference between Subjective Probability and actual statistical outcomes in game theory.
The Mathematical Proof
To understand why switching is better, we look at all possible outcomes. There are only three scenarios, determined by where the car is hidden initially.
If you always switch, the outcomes are as follows:
2. You pick Goat B ➔ Monty shows Goat A ➔ Switch to Car (Win)
3. You pick Car ➔ Monty shows a Goat ➔ Switch to Goat (Loss)
In 2 out of 3 cases, switching results in a win. The only way to lose by switching is if you were lucky enough to pick the car correctly on your very first try (a 1/3 chance).
The 50/50 Myth:
People ignore that Monty’s choice is restricted. He cannot open your door or the car door.
The Strategy:
Switching wins 66.7% of the time, while staying only wins 33.3% of the time.
Name: Source Code: Manim Implementation *
Switching doors in the Monty Hall Problem does increase your chances — it’s a great reminder that in life, being willing to change your approach or take a second look can often lead to better outcomes, even when it feels counterintuitive at first.