Do you trust your gut? A probabilistic approach.

Having some conversations about the Monty Hall experiment and got quite surprised on how many people even after the explanation, still think that they would have more chances of winning by sticking with the first choice. I started to think, there has to be a way that I can explain this better. Elaborating on this that let me to this analysis that I present to you today.

Which of the doors do you choose ?

Let me give you an introducton about what this problem is all about. Monty Hall was a game show host, in this problem at hand we analyse the part “Let’s make a deal”. Contestants were asked to pick one out of three doors. Behind one of the doors, it was a prize, and behind the others a goat. The game worked in this way, it was asked to the contestant which of the doors did they choose. After the fact, it was showed that behind one of the doors there was a goat. Now we have only two doors, the one that had been picked, and the other that wasn’t open. In this part of the game he ask the question. Do you want to change doors? Based on that, is in the best interest of the contestant to switch doors?

3 options and just one prize

Let’s look at the first facts, in the inicial phase, you have 3 options and just one prize to catch, so you have 1/3 chances of winning. You can prove mathematically that if the contestant switches the door after the fact, he now have 2/3 chances of winning.

But how do we go about proving this?

I didn’t follow the math path as my intention was to give some intuition on the problem at hand. The best way I could show that switching is in fact in the best interest, was to stop with all those peoples that I’ve talked and make this experiment. Stand by 3 doors, put a prize behind one of them, nothing in the others and following the game. I could do this 20 times, the first 10 times we wouldn’t change doors, the next 10 we would. In that way I could show that it’s really in the best interest to switch doors. But I bet you’re also thinking what I did, there’s no way in hell that a single person would’ve given me the time and pacience to make this experiment. Also if we just had made 20 times this game, we wouldn’t be so sure of the final results. So what’s the other option? Simulations!

Monte Carlo simulations to the rescue!

I ran a simulation for 10.000 games and those are the results. Just what it was mathematically expected. If you stick by your door, you have about 1/3 chance of winning, if you’ve switched, it was about 2/3.

I Still trust my gut!

Ok, ok, for the ones that still are not true believers of the wonders of probability, I’ll still try to convice you. I created another experiment called 10 cards experiment, it’s pretty like the door problem. I have 10 cards, numered from 1 up to 10. I’ll give you 10 bucks if you can choose the number 8 card. In the first place, you take a card and don’t look at it, then I’ll turn 8 cards up to you, none of these are the 8 card. Now the game is on, do you, or don’t you switch your card? In the first choice you had 1/10 chances of winning, if you switch in the last move, you’ll have 9/10 chances of winning! And here’s the simulation to support just that

Let’s exaggerate the problem!

Now let’s suppose I couldn’t convice you so far, and suppose you are still reading up to this point. Let’s go crazy and exaggerate the problem. Now I have a deck of 20 cards, or 30 cards, 50? And what about 70? 100? I created a simulation from 3 cards up to 100 and here are the results, About 50 cards you pretty much win every time if you switch your card and lose if you don’t! Crazy? No, just probability. Hope you’ve liked it, if you want more content like this, I have a YouTube Channel

Jhonatan da Silva

Jhonatan da Silva is the founder of Jhonatan da silva website and YouTube channel. When he isn’t playing around with new ways to create awesome tutorials on AI, he likes to listen and read books. He also likes to play around in Kaggle, you can look it up here