Monty Hall Paradox

[Woodrose]

I scrolled up in my Facebook page then suddenly saw this video about gambling film. Out of curiosity, I search if black jack is beatable and idea to win in Gambling.

So I tried to research something till I found out the Monty Hall Paradox. Basically, this paradox was came up right after the American TV show named "Let's make a deal".



The dealer or host will give the player 3 choices which the player have the chance to win a cash. Among the three choices, only one contains the jackpot prize and the others are just distractors.

Now here is the idea of Monty Hall Paradox, let's say you choose the option 1. The host will open the option 3 and it contains the distractor (0). After that, the host will tell you that you have the option to change your number. Will you stay in option 1 or will you change in option 2?

Mathematically speaking, you have the 33.3% chance to win in the game show when you selected the option 1. When the host opened the distractor (option 3), your odds are added. Now, from 33.3% to 50% chance to win.

And since he offered you to change your number, you have the 66.6% chance to win if you change it.

That is how the Monty Hall Paradox was explained. It talked about variable change. Once you change your number, you have the better odds to win the game. As we all know, numbers will never lie.

References:

https://www.psychologytoday.com/us/blog/machiavellians-gulling-the-rubes/201805/the-monty-hall-paradox

Here is the gambling film that I saw also:

https://fb.watch/hyCbDwFgmL/

Now here is the argument, will stay or will you change your number?

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[PawGo]

I scrolled up in my Facebook page then suddenly saw this video about gambling film. Out of curiosity, I search if black jack is beatable and idea to win in Gambling.

So I tried to research something till I found out the Monty Hall Paradox. Basically, this paradox was came up right after the American TV show named "Let's make a deal".

Yes, I know that paradox. In other words, it is based on fact that you have 2/3 chances of selecting 'wrong' box, and then the change is profitable, as one of 'wrong' boxes are eliminated. Tricky to understand at the beginning, but it works
And it is maybe not the matter of your chances 'at that moment', but that in 66% of situations the chance makes sense.

[o_e_l_e_o]

If you are finding this difficult to grasp, the concept becomes much easier as soon as you work through the three possibilities.

Let the three doors be named A, B, and C. Doors A and B contain nothing; Door C contains the prize.

Scenario 1
I pick Door A at random. The host then opens Door B and shows nothing behind it. If I switch to Door C, I win.

Scenario 2
I pick Door B at random. The host then opens Door A and shows nothing behind it. If I switch to Door C, I win.

Scenario 3
I pick Door C at random. The host then opens either Door A or Door B and shows nothing behind it. If I switch to the other door, I lose.

There are three possible scenario, and I win by switching door in two of them. Therefore, I have a 66.666...% chance of winning by switching door.

[Woodrose]

If you are finding this difficult to grasp, the concept becomes much easier as soon as you work through the three possibilities.

Let the three doors be named A, B, and C. Doors A and B contain nothing; Door C contains the prize.

Scenario 1
I pick Door A at random. The host then opens Door B and shows nothing behind it. If I switch to Door C, I win.

Scenario 2
I pick Door B at random. The host then opens Door A and shows nothing behind it. If I switch to Door C, I win.

Scenario 3
I pick Door C at random. The host then opens either Door A or Door B and shows nothing behind it. If I switch to the other door, I lose.

There are three possible scenario, and I win by switching door in two of them. Therefore, I have a 66.666...% chance of winning by switching door.

Probably, it is not that hard to grasp. But the main problem here is that would you based your decision in logic or emotion? Because if Monty Hall paradox is really working, I think it is worth to gamble your money.

Anyway, I don't know where can we use this in gambling. I used to play in color game online. And I always lose because of sticking in just 1 color. Well, I think that's a different story.

[o_e_l_e_o]

But the main problem here is that would you based your decision in logic or emotion? Because if Monty Hall paradox is really working, I think it is worth to gamble your money.

Emotions are irrelevant. The math is clear. Switching door gives you the higher odds of winning. Using emotion over logic is how you end up losing and chasing your losses.

Anyway, I don't know where can we use this in gambling. I used to play in color game online. And I always lose because of sticking in just 1 color. Well, I think that's a different story.

That's a different problem entirely. If the outcome of a draw is truly random (and provably random), then it makes absolutely no difference if you stick with the same color for every game or switch to a different color every game. It also does not matter what came before. It doesn't matter if one color has just won 20 times in a row or just lost 20 times in a row - there is the exact same chance of the next result being that color.

[Don Pedro Dinero]

I think these kinds of mental experiments are good for entertainment, but they won't happen in real life. As part of a training in statistics, fine too.

I didn't quite understand the example, but if the question is that the door that is going to open contains nothing and seeing the scenarios that o_e_l_e_o puts, it is clear that there are different odds, something that is not seen at first glance.

Emotions are irrelevant. The math is clear. Switching door gives you the higher odds of winning. Using emotion over logic is how you end up losing and chasing your losses.

Do you gamble at all? I think it's the first time I've seen you in this section of the forum and I'm sure you're not like the average user of the section who thinks that using a variant of Martingale will make him money, only to open a thread some time later saying that he's lost too much.

[panganib999]

I think you've seen a snippet of the movie 21, a cinematic retelling of the real-life story of the MIT Blackjack Team. Wwll, going back to the question, knowing how variable change works, I'd rather choose to switch than retain. I'd be more than happy to take that extra 33.3% increase in odds. Anyhows, Variable Changes doesn't just exist in gambling. In employment for instance, it is always much wiser to choose to apply for a new company than sticking to your current one in the hopes that your loyalty will be rewarded by a salary increase or whatnot. Variable change exists everywhere.

[o_e_l_e_o]

I didn't quite understand the example, but if the question is that the door that is going to open contains nothing and seeing the scenarios that o_e_l_e_o puts, it is clear that there are different odds, something that is not seen at first glance.

The key point to remember is that the host must open a door with nothing behind it. If you have also picked a door with nothing behind it, which you have a 2/3 chance of doing, then the remaining door must have the prize. Only if you have picked the prize door at the start, which you have 1/3 chance of doing, will you switch to a door with nothing. So on odds alone, it is far better to switch than it is to stick.

Do you gamble at all? I think it's the first time I've seen you in this section of the forum

A little, but not much. I mostly just talk about math and statistics in this board. There are lots of people who don't understand why Martingale is a losing strategy or fall victim to the gambler's fallacy, for example.

[bitbollo]

this example refers to Bayes' theorem/law. years ago I met some of the most important Italian reserchers on this topic. it has wide application in medical statistics (oncology).
I think in gambling it can help but considered the house edge, the casino will always be ahead.

[Merit.s]

If you are finding this difficult to grasp, the concept becomes much easier as soon as you work through the three possibilities.

Let the three doors be named A, B, and C. Doors A and B contain nothing; Door C contains the prize.

Scenario 1
I pick Door A at random. The host then opens Door B and shows nothing behind it. If I switch to Door C, I win.

Scenario 2
I pick Door B at random. The host then opens Door A and shows nothing behind it. If I switch to Door C, I win.

Scenario 3
I pick Door C at random. The host then opens either Door A or Door B and shows nothing behind it. If I switch to the other door, I lose.

There are three possible scenario, and I win by switching door in two of them. Therefore, I have a 66.666...% chance of winning by switching door.

Thanks for the proper explanation on the topic
This is an advantage for the player as he has 66.6% chance of winning after one of the doors out of three has been opened. It has more luck for players to win and the concept is a nice one too,because it is more easy to pick your choice correctly when it is only two doors,either to still choose the door you chose before or change if it was the right one.

[nullama]

I remember reading about this a long time ago, it's a great paradox.

Some people say that there's a 50% chance to win, because there's only two doors left, one with prize one without. But that's simply wrong.

It's always better to switch door to maximize your probability of winning.

A simple way of thinking of it is imagining say a million doors instead of just 3.

You pick one door in a million, and then the host opens 999,998 other doors showing no prizes inside them.

The chances that you picked the one with a prize is very low, so of course you should pick the other one, because the host knows where the prize is.

Opening the doors changes the probabilities, it's Bayesian Probability.

[Casdinyard]

Of course, having watched that film and my eyes opened to the wonders of variable change, in the event that I find myself in a situation where I had to choose between three choices and be given a chance to switch I would take it. The common bias our brain insinuates to us is that with the revelation of the second goat box, you have yourself a 50% chance at winning the game. But that is just not the case, considering how the boxes where revealed AFTER you have made the choice. To switch to the other unrevealed box will really amp your chances of winning because that is another 33.3% chance added to your odds by picking that other box. I hope this clears up the confusion on how this problem works.

[Blawpaw]

This is quite interesting. Thank you for sharing as I did not know about this. As for the movie, I have never watched it, but I guess I already have what to do tonight!
Thanks

[digaran]

Now a riddle for you to solve, imagine there are 2 doors, one leading to a grand prize and the other leading to nothing, there are 2 guards, one always tell the truth and one always lie, you have only one question to ask from one of them, what would be the question?

[o_e_l_e_o]

Now a riddle for you to solve, imagine there are 2 doors, one leading to a grand prize and the other leading to nothing, there are 2 guards, one always tell the truth and one always lie, you have only one question to ask from one of them, what would be the question?

This is a simple riddle if you think through it logically.

You can only ask one guard one question, but you have two variables - whether the guard is lying or telling the truth, and which door is the correct one. Just as you cannot solve a single equation with two unknowns, you cannot resolve both variables with a single question. So how do you remove one of the variables? You cannot remove the variable of which door is the correct one, since that is the variable you want to find out the answer to. So you have to remove the variable of which guard is telling the truth and which one is lying. How do you do that? You ask one guard what the other guard would answer. You know the answer will always be a lie.

For example:

You ask one guard how the other guard would answer the question "Is the sky blue?" If you've asked the truthful guard, he would truthfully answer that the lying guard would lie and say "No". If you've asked the lying guard, he would lie and tell you that the truthful guard would say "No". The answer will always be a lie.

So now you can ask either guard if the other guard would say the first door leads to the prize. The answer will always be a lie. So if the answer is no, you take the first door. If it is yes, you take the second door.