# Would you switch?

in LeoFinance4 months ago

You are presented with three doors, one has one million dollars behind it, two have a penny. After making your choose, one of the doors with a penny behind it will be opened and you have the option to switch doors. Do you switch?

This is an interesting puzzle that has baffled many people, including some extremely smart people. I came across the puzzle recently and I find it interesting how it either makes total sense to you, or you believe it is complete bullshit.

The fact is, you have double the chances of winning the 1 million dollars if you switch doors once you are told one of the doors that has a penny behind it. You have a two out of three chance to win the million dollars if you switch doors every time while only having a one out of three chance if you stick with your first door.

One way to look at it is like this, you have a one in three chance of picking the money on your first try. You have two out of three chances of picking the penny on the first try. There are now two doors left once you have chosen, one of these is revealed to be a penny. Chances are you see the odds of the last door being a penny as well is the same odds as it being the million dollars.

There are many ways to explain why this is not the case. The one I found most useful is this.

Let's say the million dollars is behind door number one.

If you chose door number one you should stay.
If you chose door number two you should switch.
If you chose door number three you should switch.

Let's say the million dollars is behind door number two.

If you chose door number one you should switch.
If you chose door number two you should stay.
If you chose door number three you should switch.

Let's say the million dollars is behind door number three.

If you chose door number one you should switch.
If you chose door number two you should switch.
If you chose door number three you should stay.

Are you noticing a pattern? Most of the time you would be better off switching, in fact six out of nine times you would.

Where people get confused is when you pick your first door, you have a 66% chance of picking a penny, when the other penny is revealed, most believe the odds of the final door being the million dollars is only 33%. The confusing part is the fact you still chose a penny 66% of the time. This hasn't changed, so by removing a door you know for a fact is a penny, you have drastically increased your odds of choosing the money.

This challenge was on a game show called the Monty Hall show and resulted in this problem being referred to as The Monty Hall Problem. In the case of the game show, there was two goats behind the doors and a fancy car behind the last.

The problem was made famous when it was asked in Parade magazine to Marilyn vos Savant, a women with the Guinness World Record of the Highest IQ. Thousands of people responded back saying she was wrong. This resulted in many exploring it from mathematics and computer simulation, coming to the same conclusion. If you switch every time, you will win around 66% of the time.

There are many good YouTube videos on the problem, most of them will likely leave you more confused than when you started. I found this one to explain it best.

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4 months ago (edited)

This was also in the Da Vinci Code book, and the subsequent discussion resulted in many mathematicians getting it wrong.

The best way I've heard it explained is to increase the number of doors to a ridiculous level (like 100). But for the life of me I can't remember why this helps in understanding the problem. Perhaps someone else can work it out.

edit: Ok, I read down the comments and see someone else talked about this. Imagine 100 doors, with 99 having a penny and one having a million dollars. The assistant will open 98 doors with a penny behind, leaving you two doors and the option to swap. Probability suggests you should swap, as the odds of you initially picking the million dollar door is a terrible 1 in a 100. By swapping, you increase your odds to 50%.

Definitely more than 50%. Your first choice has a very high likelihood of being penny door, so the chances that the other door has 1M dollars after the reveal would also be very likely. (99%)

I like this way of explaining the problem though, it really can help with the intuition.

This is the first explanation that makes sense to me! And even then, my head is still going in circles :D

No with 100 doors you increase your odds of winning to 99%.
Since when you picked the door initially you had a 1% chance of being correct, now you have a lot of additional information.
It is not a 50% 50% due to the fact that the door you picked was 1% so the other door is 99%.

Yep, correct.

4 months ago (edited)

I would switch. Only because i read and watched the video though :)

Same here haha, didn't seem so "logical" before I read all of these mathematics facts and explanations.

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You are presented with three doors, one has one million dollars behind it, two have a penny. After making your choose, one of the doors with a penny behind it will be opened and you have the option to switch doors. Do you switch?

This is totally/absolutely obvious. The switch would show/reveal the second "penny door" and the "one million dollars" door with it. So of course I would switch.

No it would not, there is still a chance your original door has 1 million dollars behind it.

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Ah okay. Then this is not so easy. I am not sure if I would switch or not.

This is fascinating and counter intuitive at first. When in doubt I always resort to simulations which can easily prove this beyond doubt as you have mentioned.

I've seen variants of this a few times and still not got it clear in my hear. A lot of game theory and probability is not intuitive. You can run simulations over thousands of cases to see how it works out.

I always switch doors. I believe I learned about this all the way back in high school.

It was used as an example of you needing to verify and not just assume what appears to be a simple puzzle that you already have the correct answer to it without checking it.

Stuck in thoughts 🤔

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first things first that GIF, awesome - as for the rest, in the word's of Homer J "You're selling what now?' that's so simple, yet so confusing, thank you for the pickling of ones head 😆 even reading all the examples, not a clue which way i'd go - the one that really baffles the mind is the bar of gold out of a miniature hole,

Always switch. Another simple way of looking at the problem is imagine a scenario with 100 doors that gets reduced down to two. Its the same principle but on a larger scale. Most people in the 100 doors example would assume that their first guess was incorrect and would switch.

I remember when I first found out about the explanation behind this puzzle. It really fascinated me! I didn't have a lot of statistics knowledge (still don't, but I've learned a thing or two since then) and it really stuck in my head for days! Math can be truly amazing!

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yes, Gamble with math :)

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The fact is, you have double the chances of winning the 1 million dollars if you switch doors once you are told one of the doors that has a penny behind it. You have a two out of three chance to win the million dollars if you switch doors every time while only having a one out of three chance if you stick with your first door.

This hasn't been proven to be 100% true. Its just one way of looking at things.

Another way would be:

• Contestant Picks Door two out of the three choices given. This gives a 33% chance at winning \$1 Million dollars. The other two doors each also have a 33% chance at containing 1 Million Dollars or Nothing.

• The person making the offer then reveals Door number 3 has nothing in it and offers the contestant to switch doors.

• There's now two doors to choose from not three making odds of winning 50/50. The contestant doesn't gain any advantage via switching doors.

This debate was a hot topic and debated frequently at one point. The game show Lets Make A Deal hightened the debate as they offered prices in the same fashion as your example. That being said, if I was every given such an offer I would change doors because ........ why the hell not.

This hasn't been proven to be 100% true. Its just one way of looking at things.

Yet it has been proven true without a shadow of a doubt using math and computer simulations.

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4 months ago (edited)

There's still an ongoing debate about it but for the most part, it doesn't matter to me regardless as I would always pick to switch. The decision to switch is an easy one. If a person believes the unchoosen door now has 66% of winning then its an easy switch. If a person believes the two remaining doors makes the odds 50/50% of winning then switching does no harm.

There's still an ongoing debate about it but for the most part

There is an ongoing debate about almost everything, doesn't mean it's actually valid.
Many believe the Earth is flat. ¯\(ツ)

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4 months ago (edited)

Things are always changing. New proof comes up showing old solutions and ways of thinking to be wrong time and time again. Its one of the reasons why I dislike the current school system.

As for your flat Earth example, there was a time that most thought Earth was flat and the people who thought it was round were the crazy ones.

In math things never change, once something has been proven it is true, and the probability is solidly in maths territory.

What I find interesting is:

• If the door eliminated is removed by chance rather than by knowledge of the penny's location, and it doesn't result in the penny being behind it, then there's no advantage to swapping doors. It becomes 50/50 between the two remaining doors.
• So you need to know whether the person eliminating the door did so out of knowledge or out of luck in order to determine the odds of whether swapping is beneficial.

The same is true with 1 million doors:

• If someone removes the 999,998 incorrect doors through knowledge then you can be pretty much certain that swapping will give you the penny.
• But if they somehow manage to remove 999,998 doors through chance without knowledge, then there's a 50/50 chance with the two remaining doors.

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4 months ago (edited)

Your scenario cannot happen because the host is not allowed to open the chosen door (otherwise 'stay or switch' doesn't make sense). So if the contestant picks a penny door, the host is forced to open the other penny door.

Sure. In a game show that methodology cannot be applied. The host cannot open a door using chance in case they pick the winner.

But outside of a game show, say as a puzzle in a book, the idea holds that:

• If you have chosen a door
• And an alternative door is opened without success
• Then the probabilities of whether you should stick or switch depend on whether the alternative door was opened through knowledge or chance.

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Oh, I see what you are saying now. Your setup is a one-time scenario, and the contestant doesn't know the rules of the host. (e.g. the host rule was to flip the other doors randomly, and it happened to get lucky and not reveal the 1M, or the host in this case just opened the penny door and tells the contestant that it was flipped randomly).

But I think the wording of the original post is clear that "one of the doors with a penny behind it will be opened", and that implies it cannot be done by chance.

The thing that makes this old chestnut confusing and interesting is the decision on whether the odds are based on two individual events or a combined event.
It's in a similar being to the odds of tossing two heads in a row, is it 1/2 + 1/2 or 1/2 X 1/2...if you understand the maths behind that then the Monty Hall Problem should be less confusing albeit with one additional choice.

I probably wouldn't. I understand the concept but can't make it stick in reality.

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I too have been down this same rabbit hole many years ago, nice of you to bring it up again for all of us to ponder over.

I didn't understand it at first, but then I got it sorted out. So you pick your door but they don't open it. Then they open one of the ones that had a penny and then you have to choose if you stay or you switch. I get it now. I was totally misreading it at first.

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I'm feeling very much like when Sarah was made to choose between the two doors in Labyrinth.... one leads to the castle, the other one to certain death... though I was never quite sure she was wrong (spoiler she chose wrong)

Yeah the Monte-Hall problem melted my brain at first as well.
It's a great teaser for probability.

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Next you should talk about the prison dilemma scenario, it has a very similar sort of logic to it but with its own nuances. I have heard that there used to be tournaments out there to see who could develop the best strategy when competing against others and one person used game theory to discover the best strategy. Its very interesting and I apply it in life at times. Thats something you might also be interested in???

Or as a friend of mine simplifies it - ALWAYS change your mind!

I would say it would always be a 50/50 chance of picking the right door. In first try it would be 50/50 between winner pick or non-winner, even if one options has two doors left.

Once you pick the wrong door, it is still 50/50 chance to pick the winner between remaining doors. Opened door becomes irrelevant at that point.

How could it be a 50/50 chance when you are picking one out of three on the first try?
You are a Python dev, program it out, you will see it isn't 50/50 in either situation.

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4 months ago (edited)

Because ultimately everything comes down to 0 vs 1, True or False.

Especially in this case, since I can open only one door at a time, my options are:

a. Door 1 vs (Door 2 + Door 3)
b. Door 2 vs (Door 1 + Door 3)
c. Door 3 vs (Door 2 + Door 3)

Whichever I choose, I face with 50/50 chance.

The only difference is, with the first try I have 50% chance to get the correct one, and 50% to get another opportunity to try. With the second try, I have 50% chance to get the correct one, and 50% to lose.

But also, coins only have two sides. I would toss a coin to decide for me. And for that reason I need to make it 50/50.

P.S. I misunderstood the puzzle at first. But I would still use the coin toss, and treat each step as 50/50 chance. So can't tell if I would switch or stay until I flip the coin.

If someone came along to fuck with me with this psychotic experiment, I'd simply screw up until the doors are open, walk up and take the money, then tell them that's what they get for being so smart.

Dwayne Johnson is behind the million dollar door protecting the money and nullifying your plan. Sorry mate, but have a penny!

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He's an actor. I'm not. Now what?

Really? That "puzzle" is old as the world, everybody knows that and it obtains 657+ upvotes and 100 dollars?
I did in past posts about differential geometry, number theory, algebra and other hard math stuff which were sometimes earning less than 1 dollar, and such sorry to say that "nothing" earns 100 dollars!
Maybe next time I will do post about how to win in Tic Tac Toe.

You should, it would probably be better than this.

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You know nothing about history behind that post. The day before I wrote that post my mother has been diagnosed with brain cancer - and it would have been 4th cancer in her life - a few days later it appeared that the diagnose was not proper.
If You look at my wallet You see that I got rid of Hive and HBD for that post - I supported teenager from Venezuela, country which has very big problems. I behaved fair.

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Interesting!

Bagus nih

This game is awesome and in another way it is a puzzle. I have heard of this kind of game before but to me, I have to exercise patience and stay in the first choice .

Wao this is so technical. I will just pray to God and just try an use one mind to choose

I already knew you always have to switch, but kind of forgot the logic behind it lol Thanks for reminding me.