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Frank Hovis

Puzzles, to help with lockdown

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Any and every.

I'll start with one that has been on here before but is one that you really have to think through to understand as against just knowing the answer.

New posters only because old hands have seen it. Unless you are an old hand and it baffled you the first time.

The Three Door Prize

In the final round of a quiz show there is a prize behind one of three doors A, B and C.  There is an equal chance of the prize being behind any door.

You will pick a door and then the host will open one of the other two doors to show that the prize is not behind it.

You then have the option of staying with your original choice or swapping to the remaining closed door.

Should you stick or swap and why?

Less verbosely:

Doors A B C, equal chance behind any

You pick A

Host opens B to show it's empty

Stay with A or swap to C?

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2 minutes ago, GBDamo said:

I hate probability cos I can't fathom it.

I'd stick purely out of the fear of having the prize but losing it byswapping.

Yes, regret is a powerful motivator.

I have spent a lot of money that I would rather not have spent on works' lottery syndicates because however unlikely was a big win I didn't want to be the only one who didn't get it.

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5 minutes ago, Option5 said:

Can I ask Monty Hall?

I knew it was a real quiz show but couldn't recall it beyond being a US show.

Even if you know the answer do you genuinely understand it?

I was kicking it around on my walk this morning refining the logic down to its simplest form at which point it becomes obvious.

(In the garden at the moment, it sounds like I've been locked in an aviary with all the fluttering and singing. Nice though.)

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2 minutes ago, Option5 said:

Switch because you first choice had a one in three chance, if you stick it's odds don't change. However if you switch it's a new game and the odds are one in two so that's the better choice.

Bollocks I know but that's the way it's done.

This. Was explained in the film 21. Also as I recall there is a website where you can have a go, if I remember it is better to switch.

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5 minutes ago, Option5 said:

Switch because you first choice had a one in three chance, if you stick it's odds don't change. However if you switch it's a new game and the odds are one in two so that's the better choice.

Bollocks I know but that's the way it's done.

I would express it differently.

There is a one third chance of the prize being behind doors A, B and C.

You choose A and your odds of being right are one in three.

There is however a two in three chance of it being behind doors B and C.

When the host opens B to eliminate that option then there is now a two in three chance of it being behind door C because that now combines the likelihood of it being behind either door B or door C.

So by switching to door C you are effectively picking two doors - B & C - as against your original choice of a single door.

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I have to believe mathematically it's true that you win more by switching, but I don't actually believe believe it if you see what I mean. 

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6 minutes ago, Frank Hovis said:

I would express it differently.

There is a one third chance of the prize being behind doors A, B and C.

You choose A and your odds of being right are one in three.

There is however a two in three chance of it being behind doors B and C.

When the host opens B to eliminate that option then there is now a two in three chance of it being behind door C because that now combines the likelihood of it being behind either door B or door C.

So by switching to door C you are effectively picking two doors - B & C - as against your original choice of a single door.

Actually you explanation gives different odds and is probably better.

My answer is that the best odds are 1 in 2, your explanation gives best odds of 2 in 3. A 50% chance of winning against a 33.3 % chance.

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1 minute ago, Funn3r said:

I have to believe mathematically it's true that you win more by switching, but I don't actually believe believe it if you see what I mean. 

You're switching from picking one door to picking two doors.

The host isn't tricking you; he's showing which door to pick to double your chance.

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Don't get it.

All that's gone before has gone, surely.

It's a 50:50 that's all. It is either in B or C.

If that simulation is based on a flawed premis then it will deliver a false outcome.

 

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15 minutes ago, Great Guy said:

Some key information is that the host knows which door has the prize..... He doesnt open the door with the prize behind it....

But he cannot open the door you have picked either...

So if you have picked correctly (0.33) then he is bluffing (0.5) but if you have not picked correctly (0.66*) he isn't (0.5)

* more likely

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1 minute ago, GBDamo said:

Don't get it.

All that's gone before has gone, surely.

It's a 50:50 that's all. It is either in B or C.

If that simulation is based on a flawed premis then it will deliver a false outcome.

 

The prize can be behind A, B or C and doesn't move.

One in three chance of it being behind door A.

Two in three chance of it being behind doors B & C.

Instead of opening an empty door the host lets you switch from A to both B & C so you have double the chance.

The key is that his opening the empty door doesn't matter; you know that even if it is behind B & C it will only be behind one of them.

 

 

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1 minute ago, Frank Hovis said:

The prize can be behind A, B or C and doesn't move.

One in three chance of it being behind door A.

Two in three chance of it being behind doors B & C.

Instead of opening an empty door the host lets you switch from A to both B & C so you have double the chance.

The key is that his opening the empty door doesn't matter; you know that even if it is behind B & C it will only be behind one of them.

 

 

All this A, B, C is what WAS.

What IS is the prize is behind B or C and you are given a choice of picking one or the other; 50:50.

This whole thing conflates two sets of choices and probabilities to obscure the obvous.

I can't help but think there is an emperor's new clothes element to this.

 

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Just now, GBDamo said:

All this A, B, C is what WAS.

What IS is the prize is behind B or C and you are given a choice of picking one or the other; 50:50.

This whole thing conflates two sets of choices and probabilities to obscure the obvous.

I can't help but think there is an emperor's new clothes element to this.

 

 

The host's opening the door is there to distract you.

Whilst it appears to tell you something it actually adds no new information.  You know at the start that at least one of doors B & C has nothing behind it.

Pretend he hasn't done it because it is a distraction and changes nothing.

Spin back to the beginning.

You have two choices albeit stepped in time.

You can pick door A.

Or you can pick doors B & C.

You know one of B & C has nothing behind it. You knew that at the beginning.

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30 minutes ago, Funn3r said:

I have to believe mathematically it's true that you win more by switching, but I don't actually believe believe it if you see what I mean. 

Let's do it a different way that is often more intuitive...

Imagine you don't have 3 doors, but a million. 

You pick a door, the host then opens all of the remaining doors except 1. Do you stick or change?

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What if... 

You get a flat tyre on the way to the quiz show so you arrive late. They have already opened one of the bad doors let's say A. 

Now you have to choose between B or C. Is there still some sort of advantage in not selecting the door you felt like selecting? 

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2 minutes ago, Roger_Mellie said:

Here's one that often gets pulses racing:

A mother has 2 children, one is a boy, what is the probability that his sibling is female?

Is this a trick one as well... 

How many genders are there? 

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