Rng Primer: Monty Hall Problem
Hello everybody, Geek Generation here.
This post is the follow up to the posts Rng Primer: Gambler’s Fallacy and Coding Luck into a Game. In my last post, i may have put it a little too strongly by saying that “we don’t understand the math behind”. What i meant to say is that even with knowledge of the math behind, we may still make mistake and apply the math wrongly.
Monty Hall Problem
Imagine you’re in a contest and there are three doors. The prize is behind one of the doors while the other two doors open into an empty room. Let’s say the doors are named A, B and C. Not knowing which door opens to what, you make a choice selecting one of the doors, door A.
But the chosen door is not opened immediately. Instead, the game host opens one of the remaining doors, door B, to show that it is empty. The game host then asks if you would like to change to door C or hold on to door A.
Some might think that the prize has 50% chance of being behind either of the remaining doors and hence it doesn’t matter whether door A or C is chosen. But that is incorrect. At this point, swapping to door C actually gives a better probability of winning.
This is so counter-intuitive that many people couldn’t agree with it. That is until they perform the simulations themselves and found that the probability was indeed higher when the swap was made.
I find that the simplest way to explain the Monty Hall Problem is this; instead of 3 doors, imagine 1,000,000 doors. You pick one door and the game host then opens 999,998 doors. Do you make the swap in this scenario? Of course you would. The door you first chose only had one in a million chance of being correct. But the other door has a probability so much higher than 50%.
The reason why i talk about Monty Hall Problem is to highlight that it is entirely possible that the correct math is counter-intuitive to us. That we might have trouble relating to or even interpreting probabilities. That our perception of likelihood is often fundamentally flawed because of our inability to accurately judge and estimate probabilities.
Confounding this problem of interpreting probabilities is our own personal risk adversity and the expected value of the probability.
How likely to happen is the probability 1/6? The question, as it is, is hard to answer.
If you paid $10 roll a 6 sided die, and would win $11 if you rolled a 6, the answer would be 1/6 is not very likely to happen. If instead, you would win a million dollars, then the answer would be that 1/6 is very likely to happen.
Of course, given such exact values and probability, we could easily calculate expected value. ((1/6) x 11) – 10 or ((1/6) x 1000000) – 10. But the fact was that our perception of likelihood of things happening is changed by factors other than the probability itself, which has remained 1/6 in both scenarios.
In alot of other real world problems, the factors are less tangible and expected value and personal risk adversity usually cannot be tagged by numerical values.
Ok so much for now, Geek Generation out.