People are inherently bad at probabilistic thinking.
Our intuition deceives us. We often encounter seemingly contradictory phenomena that go against our expectations.
The most famous example is the Monty Hall paradox. Let's see what it is and how to resolve it!
↓ A thread. ↓
Our intuition deceives us. We often encounter seemingly contradictory phenomena that go against our expectations.
The most famous example is the Monty Hall paradox. Let's see what it is and how to resolve it!
↓ A thread. ↓
In the ’60s, there was a TV show in the United States called Let’s Make a Deal.
As a contestant, you faced three closed doors, one having a car behind it (that you could take home), while the rest were empty.
You had the opportunity to open one.
As a contestant, you faced three closed doors, one having a car behind it (that you could take home), while the rest were empty.
You had the opportunity to open one.
Suppose that after selecting door No. 1, Monty Hall, the show host, opens the third, showing that it was not the winning one.
Now, you have the opportunity to change your mind and open door No. 2 instead of the first one.
Do you take it?
Now, you have the opportunity to change your mind and open door No. 2 instead of the first one.
Do you take it?
At first glance, your chances are 50%-50%, so you are not better off switching.
However, this couldn't be further from the truth!
Let's explain why.
However, this couldn't be further from the truth!
Let's explain why.
To set things straight, let’s do a careful probabilistic analysis!
Let Aᵢ denote the event that the price is behind the i-th door, while Bᵢ is the event of Monty opening the i-th door.
Regarding the price, each door is equally probable.
Let Aᵢ denote the event that the price is behind the i-th door, while Bᵢ is the event of Monty opening the i-th door.
Regarding the price, each door is equally probable.
However, Monty opening the third door changes everything.
When building probabilistic models, we must account for events that influence each other's occurrence.
This is formalized by conditional probability. (Below is my explainer.)
When building probabilistic models, we must account for events that influence each other's occurrence.
This is formalized by conditional probability. (Below is my explainer.)
https://twitter.com/TivadarDanka/status/1492127542068666376
We want the probability of the price behind the first and the second door, given that Monty Hall opened the third one.
By thinking from the perspective of the show host, which door would you open?
By thinking from the perspective of the show host, which door would you open?
If Monty knows that the price is behind the 1st door, he opens the 2nd and 3rd one with equal probability.
However, if the price is actually behind the 2nd door (and the contestant selected the 1st one), Monty always opens the 3rd one!
However, if the price is actually behind the 2nd door (and the contestant selected the 1st one), Monty always opens the 3rd one!
Using the Bayes formula, we can finally calculate the probabilities we need.
In summary, if we picked the 1st door and Monty opened the 3rd one, we are twice as likely to win if we switch doors!
What seemed counterintuitive is now proven.
In summary, if we picked the 1st door and Monty opened the 3rd one, we are twice as likely to win if we switch doors!
What seemed counterintuitive is now proven.
(If you are not familiar with the Bayes formula, here is an explanation thread I posted recently. Check it out!)
https://twitter.com/TivadarDanka/status/1481254896137506816
Having a deep understanding of math will make you a better engineer. I want to help you with this, so I am writing a comprehensive book about the subject.
If you are interested in the details and beauties of mathematics, check out the early access!
tivadardanka.com/book
If you are interested in the details and beauties of mathematics, check out the early access!
tivadardanka.com/book
I regularly post deep-dive explanations about seemingly complex concepts from mathematics and machine learning.
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Mathematics is beautiful, and I want to show this to you.
If you have enjoyed this thread, consider giving it a retweet and following me!
Mathematics is beautiful, and I want to show this to you.
I have made a small mistake in this thread, pointed out by @AlfonAmayuelas!
The crux of the issue is that P(B1), P(B2), and P(B3) are unknown to us. Therefore, the final probabilities are obtained the following way. ↓
The crux of the issue is that P(B1), P(B2), and P(B3) are unknown to us. Therefore, the final probabilities are obtained the following way. ↓
https://twitter.com/AlfonAmayuelas/status/1493590569414713344
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