What are the odds of winning?
 

So, this has been circling the Internet for years and was once the concept of a gameshow in the U.S.


The question is, what do you pick? What are your chances?












Many people say it is not 50/50 and go into long explanations about it.
In my opinion, once you have been shown the booby prize in one of the doors and you picked the other door, then your chance of winning is 50/50 as there is your door you selected and the other door. So it is in either yours or the other one.






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Re: What are the odds of winning?
 

https://en.m.wikipedia.org/wiki/Monty_Hall_problem

Re: What are the odds of winning?
 

That Wiki explanation is a bit long-winded. Easier to say "the chance of you picking the car to start with is 1/3. So the chance of it remaining behind another door is 2/3. It's not the door the host opens, so the 2/3 chance must be the other door".

If you don't believe it, try it. You'll soon find they are correct.

---------- Post added at 14:19 ---------- Previous post was at 14:00 ----------

ps. If you want some software to help you, here's some we prepared earlier:
https://www.cableforum.uk/board/show...9#post34854319

Re: What are the odds of winning?
 

The 1st round is immaterial, as you still don't know whether your 1st choice was right or wrong. There is a 50/50 chance of selecting the right one. Doesn't matter whether you stick with or swap your original choice.

There is no win or lose event for the 1st round, so you can't multiply the two probabilities together. Doesn't matter which one of the 3 you pick in the first round, as you are starting again with just 2 choices.

If it was 2 rounds with 2 different people separately and independently, making the selection and both had to be right, then yes you can multiply the probabilities together to get a 1 in 6 chance of both people being right.

Re: What are the odds of winning?
 

At the point of your final choice, there are two doors, and you dont know which one has the car, so your odds are 50/50. How can they possibly be anything else ?

Re: What are the odds of winning?
 

Because it's more likely to be behind one door than the other.

If you don't believe it, make 3 cards (2 goats and a car), ask a friend to be the host, and play a few games. Or write yourself a computer simulation.

Re: What are the odds of winning?
 

You could have a million doors in the 1st round. Doesn't matter whether you pick the correct one or one of the 999,999 incorrect ones. The result is the same, you then get to pick from just 2 doors. Any of the 1m 1st round choices is "correct".

Re: What are the odds of winning?
 

But you don’t…

Re: What are the odds of winning?
 

:confused:

Re: What are the odds of winning?
 

The whole point is that there are three doors initially, and it’s about the mathematical odds…

Re: What are the odds of winning?
 

Doesn't change the odds. If you choose the right one in the 1st round, you don't win. You don't lose either.

If it was the situation that if you guessed correctly in the 1st round, you would win, then EVERYBODY would win. As if you lost in the 1st round, you would know that it definitely was the one of the remaining 2, you didn't pick 1st time around.

Re: What are the odds of winning?
 

Ok, let's play with a million boxes. That's 999,999 goats and a car.

You pick a box. The chance that you have picked the car is 1 in a million.

So the chance that the car is in one of the other boxes is 999,999 in a million.

The host now opens 999,998 boxes that he knows contain a goat. So the chance that the car is in the one remaining box is 999,999 in a million.

So do you want to keep your original pick (1 in a million), or do you want to swap (999,999 in a million)? Do you like goats?

Re: What are the odds of winning?
 

I dont think so.

You have two doors, and no idea which is the correct one, your chances are equal, how can one possibly be more likely then the other ?

How is that relevant ?
Again, you are left with two doors, and you dont know which one is right, how are your odds at that point anything other than 50/50 ?

---------- Post added at 21:16 ---------- Previous post was at 21:10 ----------

Again, how is this "the key" or relevant ?
Only the host knows where the car is, you have no clue at all, as far as you are ever concerned, its behind one door other the other, you have no further information at all to help you decide, so your odds of being right when asked to choose again are 50/50.

Re: What are the odds of winning?
 

Doesn't matter if you get your 1st round choice correct, you don't win at that stage.
There is always just the one other door.

Re: What are the odds of winning?
 

I told my sister this one years ago and she gave exactly your reply. You're both wrong. Years ago a maths professor in the US explained it and she was ridiculed by many of her peers until she proved it mathematically. Mathematical odds aren't always intuitive.

Re: What are the odds of winning?
 

You CAN'T win the 1st round. There is NO outcome. Win or lose, you get a chance with just 2 doors.
No mention of what if you got it right 1st time.
3 doors, A, B, & C. Prize is behind A.
If you can win 1st time around, then if you pick B, the host will open C. Pick C and B is opened. Either way, you would then know it was A and would win. That is why you can't win 1st time around.


There are always n-2 booby doors to open.


If the host opened one of the other doors at random, then yes the 1st round choice would matter. If you didn't make the right 1st choice, then there is the risk the host would open the prize door instead. But the host knows where the prize is, so it can't be at random.

Re: What are the odds of winning?
 

But I do have an idea, because the host has shown me a box that definitely contains a goat.

Yes there is, and you should pick it.

Re: What are the odds of winning?
 

No you dont have any idea at all.
Knowing whats behind door number 3 gives zero clue as to whats behind door 1 or door 2. Pick one, its 50/50.

You could have 10 doors, and be told whats behind 8 of them, that still doesnt tell you whats behind the other two, you still have a 50/50 chance of picking the right one out of the remaining two.

Re: What are the odds of winning?
 

Seems this puzzle is still keeping us guessing. I like that.


Everyone seems to keep telling me I'm wrong but I see each round as individual rounds.
It doesnt matter what you started with as by the time of your final choice you will only have two doors left.
Both of these remaining doors could have a goat or car behind them. It is 50/50


You will leave with one or the other.


So what if the host has showed me a door has a goat behind it? I still have no idea what is behind the other two.

Re: What are the odds of winning?
 

And therein lies the problem. You can’t accurately calculate a probability if you don’t know what it is you’re calculating the probability of. In this case the rules of the game are unclear (whether there are ‘rounds’, how each round is played and whether things done in one round affect the next).

It is possible to produce a definitive answer, but only if there’s a definitive question.

Re: What are the odds of winning?
 

Is there an inaccuracy in the description of the problem?
As the host knows where the prize is, they can only open all but one of the booby prize doors. They can't ever open the prize door. It would be unfair.
It's can't be a real type of game show scenario, and that is where the confusion lies. As there is no win or lose in the 1st round, you add the win options and the lose options together, giving 3 in 3 in odds for that 1st round.

Re: What are the odds of winning?
 

You have three doors to pick from. Suppose you pick door A, there is a 1/3 chance of it having the prize. So there is a 2/3 chance of it being behind B or C. If the host then tells you it isn't behind door B there is a 2/3 chance of it being behind C. So double the chance of that of it being behind A. Rather counter intuitive I know but numbers can be weird.

Re: What are the odds of winning?
 

As a game show it's pretty rubbish. Just an interesting thought experiment.

There again. this one's even simpler, yet brilliant:
https://www.youtube.com/watch?v=0UGuPvrsG3E

Re: What are the odds of winning?
 

The definitive question is, should you stay with your first choice or swap to the other unknown.

Re: What are the odds of winning?
 

So it’s 50/50 then? 👍

Re: What are the odds of winning?
 

If you like :D

Re: What are the odds of winning?
 

Or maybe a more worthwhile question to ask would be:


With petrol and electricity prices on the constant rise, Would it be better to win the goat!

Re: What are the odds of winning?
 

I mentioned this to a friend of mine who teaches maths.

His reply (word for word) was "ah, that old chessnut, always change your mind" [ ie switch your choice ].

So I asked him to explain.

He said its a poor example to be able to understand, its far easier to understand how it works with a more extreme example.
The important points are you made you initial choice when all options were available, but you second choice is after the [wrong] options have been reduced.

His example ;

Take a pack of cards, and spread them all out on a glass table.
Then make you choice for which you think is the Ace of Spades (AOS).
Then get a friend to look under the table, and tell you 50 wrong cards to remove (but not yours).

This leaves you with your choice, and one other card.
However, the odds are not even, the chances the other card is the AOS are huge.

The odds you initially chose the AOS correctly is 1 in 52 [low], so the odds of it being in the other cards is 51 in 52 [high].
Now your friend has removed 50 incorrect cards, the chances of the remaining card being the AOS are still 51 in 52, so always change your mind, and pick that card.

It would only be 50/50 if your friend had removed the 50 cards before you made your first choice.

I understand now. :D

Re: What are the odds of winning?
 

The 1st pick could be AOS or any of the other 51 cards, makes no difference to the outcome. 2 cards will always be left, the AOS and another card.

Any of the 52 cars is a "winner", as there's no losing choice. It's a complete redo with 2 cards instead of 52.
As a game show idea the 1st round would be meaningless. You could just go straight to the 2nd round.

Re: What are the odds of winning?
 

But it's far more likely to be the card you didn't pick than the one that you did. That's why you should always swap. Get yourself a pack of cards and try it.

But then you wouldn't have a card to swap. :confused:

Re: What are the odds of winning?
 

With the 3 doors scenario, if you pick correctly first time, then the host has TWO doors to pick from. If you pick one of the 2 the wrong ones, then only ONE to pick from. Overall there are 4 possible end results.
There is a prize door and your initial choice of door. There is a 50/50 chance they are one in the same. That is the question you are being asked, no matter how many doors you start with.

The mistake has been made in treating all non-prize doors as a single outcome.
Prize is behind A.
Combination 1 : Pick A, Host opens B, stick wins
Combination 2 : Pick A, Host opens C, stick wins
Combination 3 : Pick B, Host opens C, swap wins
Combination 4 : Pick C, Host opens B, swap wins
If you're going to treat 1&2 as a single combination, then you have to treat 3&4 as a single combination. That still brings you around to a 50/50 chance.
Now 4 starting doors, prize is behind A
Comb 1 : Pick A, opens B&C, stick wins
Comb 2 : Pick A, opens B&D, stick wins
Comb 3 : Pick A, opens C&D, stick wins
Comb 4 : Pick B, opens C&D. swap wins
Comb 5 : Pick C, opens B&D. swap wins
Comb 6 : Pick D, opens B&C. swap wins
50/50 again.
Would you when throwing a pair of dice, treat all non-double six combinations as single combination?

Re: What are the odds of winning?
 

It's already been explained several times, please don't get lost down that rabbit hole :)

Re: What are the odds of winning?
 

But you're twice as likely to pick either B or C than you are to pick A. So swap wins twice as often.

Re: What are the odds of winning?
 

The odds are always in favour of swapping.
If you dont understand that after the example I gave, you never will.

[ and no, of course its not a real game show, its just a maths teaser, no one would run it as a real show because the odds are in favour of the participant ]