How to Win a Sportscar Using Probability
Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a sportscar. Behind the others there are only goats. You pick a door, say No. $1$, and the host, who knows what's behind the doors, opens another door, say No. $3$, which has a goat. He then says to you, "Do you want to pick door No. $2$?" Is it to your advantage to switch your choice?At first glance, this problem seems simple. But it’s one of those puzzles where our intuition completely misleads us. It’s a perfect example of how probability defies common sense. The original question became famous as a question from reader Craig F. Whitaker's letter quoted in, and solved by, Marilyn vos Savant in the Ask Marilyn column in Parade magazine in $1990$. Before we delve into the actual answer, what do you think about this question? Isn’t it just more intuitive to keep your choice? Or do we need to switch to have a better chance?
The Answer
Well, Savant's response was that the contestant should switch to the other door. If we assume that each door has an equal chance of having the sportscar, the switching strategy has a $2/3$ probability of winning the car, while the strategy of keeping the initial choice has only a $1/3$ probability.“WAIT WHAT?” Is something you may ask yourself. Well, do not worry. Many readers of Savant's column refused to believe switching is beneficial and rejected her explanation. After the problem appeared in Parade, approximately $10\,000$ readers, including nearly 1,000 with PhDs (which are supposed to be smart people 😉), wrote to the magazine, most of them calling Savant wrong. Who is in the right here?
When the player first makes their choice, there is a $2/3$ chance that the car is behind one of the doors not chosen. This probability does not change after the host reveals a goat behind one of the unchosen doors. When the host provides information about the two unchosen doors, revealing that one of them does not have the car behind it, the probability that one of the two unchosen doors initially had the sportscar was $2/3$. However, nothing really changed and the only unchosen door will have a chance $2/3$ of having the sportscar. However, keeping your original choice will just come down to the $1/3$ chance of the car being behind the door the contestant chose initially.
This argument only works when the contestant chooses their door randomly and the host then randomly chooses a door not containing the sportscar. Using these assumptions, there is more information about doors $2$ and $3$ than was available at the beginning of the game when door $1$ was chosen by the player.
When the host opens a door, he’s actually giving you information. The door he opens tells you something about where the prize isn’t, and that makes the door you didn’t pick more valuable. That’s the key insight most people miss. Switching doors isn’t the same as just picking randomly between the two doors left. When you switch, you’re using the new information the host has revealed you’re making a smarter choice based on what you’ve learned. The key idea is that the $2/3$ chance that the car wasn’t behind your original choice doesn’t vanish. It just gets concentrated on the one remaining unopened door.
Suppose there are a million doors, and you pick door #$1$. Then the host, who knows what's behind the doors and will always avoid the one with the prize, opens them all except door #$420 \, 069$. You'd switch to that door pretty fast, wouldn't you?Ok, she did not mention that specific number. But the logic is sound.
When having one million doors then your odds of picking the right door are just $1$ against $1\,000\,000$. However, the odds of all the other doors are concentrated in one massive $999\,999$ against $1\,000\,000$. So revealing all $999\,998$ doors and leaving one unopened will shift all the odds to the unpicked and unopened door. This exaggerated version helps make the logic obvious. With so many doors, it’s clear your first choice was almost certainly wrong. In this sense, the original problem with only three doors is the hardest to wrap your head around. The conclusion for the problem only considering three doors is
If you don’t switch, then you’re going to be right if and only if your initial guess was correct ($1/3$). If you do switch, then you’re going to be right if and only if your initial guess was wrong ($2/3$).
Even when given explanations, simulations, and formal mathematical proofs, many people still did not accept that switching is the best strategy. Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating Savant's predicted result. That’s the beauty of math. When logic and intuition clash, we can always turn to experiments.
Repeating the experiment many times
If you are still not convinced then maybe it is also time to consider a simulation. We will run this experiment for a large amount of times and see whether our earlier logic really holds. A random door will be picked that has the car. A random door will be picked by the player. The host will then open the unchosen door without car. Then the participant can either pick the other door or the originally chosen door. For each experiment, we register which option, switching or keeping doors, was succesful. For people interested in coding and trying this out for yourself you can always run the following Python-code:
import random
def simulate_monty_hall(n_experiments=10000):
success_switch = 0
success_keep = 0
for _ in range(n_experiments):
door_car = random.randrange(3)
door_picked = random.randrange(3)
if door_car == door_picked:
success_keep += 1
else:
success_switch += 1
print("Switching win rate:", success_switch / n_experiments)
print("Keeping win rate:", success_keep / n_experiments)
Conclusion
The Monty Hall problem is what mathematicians call a veridical paradox. That means it looks completely absurd at first. Your intuition screams that it can’t be right. Yet the math proves it is true. It’s one of those puzzles that makes you question how well you actually understand chance.In fact, this problem isn’t unique. It belongs to a fascinating family of brain-twisting puzzles where new information changes what you think you know, even though nothing physical about the situation changes. What makes these paradoxes so addictive is that they remind us how fragile our intuition can be and how mathematics can reveal truth even when our instincts look the other way.
Comments
Post a Comment