The Engineering behind paper planes
The Monty Hall Problem – A statistics problem even boggling the mind of Brooklyn 99’s arguably most smart characters, is the simplest way to explain re-evaluating your decisions as new information emerges, and how probability works on this principle.
Let’s explore how this puzzle works-
- There are 3 doors, behind which are two goats and a car.
- You pick a door (Let it be door A). You’re hoping for the car.
- Monty Hall, the game show host, examines the other doors (B & C) and opens one with a goat. (If both doors have goats, he picks randomly.)
Here’s the game: Do you stick with door A (original guess) or switch to the unopened door? Does it matter? Surprisingly, the odds aren’t 50-50. If you switch doors you’ll win 2/3 of the time!
Results of the game using different strategies-
If you use a “pick and hold” strategy, the percent win rate settles at 1 in 3 instances. But, if you use a “pick and switch” strategy, The percent win rate settles at 3 in 5 instances, which is greater than picking and holding.
Why does switching increase your win percentage?
If you pick a door and hold, you have a 1/3 chance of winning.
Your first guess is 1 in 3 since there are 3 random options
If you rigidly stick with my first choice no matter what, you can’t improve your chances to win. Monty could add 50 doors, blow the other ones up, do a voodoo rain dance — it doesn’t matter. The best chance you have with your original choice is 1 in 3. The other doors must have the rest of the chances, or 2/3. But, this explanation does not explain why odds of you winning improve as you switch. Here comes the concept of filtering. Let’s see why filtering doors helps switching to be attractive. Imagine this variant:
- There are 100 doors to pick from in the beginning
- You pick one door
- Monty looks at the 99 others, finds the goats, and opens all but 1
Do you stick with your original door (1/100), or the other door, which was filtered from 99? It’s a bit clearer: Monty is taking a set of 99 choices and improving them by removing 98 goats. When he’s done, he has the top door out of 99 for you to pick.Your decision: Do you want a random door out of 100 (initial guess) or the best door out of 99? Said another way, do you want 1 random chance or the best of 99 random chances?We’re starting to see why Monty’s actions help us. He’s letting us choose between a generic, random choice and a curated, filtered choice. Filtering is better. But, two choices usually mean a 50-50 chance, why doesn’t it happen here?
Overcoming Our Misconceptions
Assuming that “two choices means 50-50 chances” is our biggest hurdle.
Yes, two choices are equally likely when you know nothing about either choice. If you picked two random books and asked “Which is more expensive?” you’d have no guess. You pick based on the name of the book, and 50-50 is the best you can do. You know nothing about the situation.
Now, let’s say book A is a very common book, found everywhere, and Book B is a rare book which has been awarded the last 10 years in a row. Would this change your guess? Sure thing: you’ll pick Book B (with near-certainty). But, someone uninformed would still call it a 50-50 situation, Hence we conclude that Information matters.
The more you know…
Here’s the general idea: The more you know, the better your decision.
With the books, you know more than your friend and have better chances. Yes, yes, there’s a chance the common book can also win awards, but we’re talking probabilities here. The more you test the old standard, the less likely the new choice beats it.This is what happens with the 100 door game. Your first pick is a random door (1/100) and your other choice is the champion that beats out 99 other doors (aka the MVP of the league). The odds are the champ is better than the new door, too.
Here’s the key points to understanding the Monty Hall puzzle:
- Two choices are 50-50 when you know nothing about them
- Monty helps us by “filtering” the bad choices on the other side. It’s a choice of a random guess and the “Champ door” that’s the best on the other side.
- In general, more information means you re-evaluate your choices
The fatal flaw in the Monty Hall paradox is not taking Monty’s filtering into account, thinking the chances are the same before and after. But the goal isn’t to understand this puzzle — it’s to realize how subsequent actions & information challenge previous decisions.