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Puzzles and Conundrums


Lefty

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Guest LaBambaBar

Looks like we agree on something.  :biggrin:

 

After the door with the goat is opened their is now 1 door left with a car and 1 door with left with a goat.

 

If you had the car initially and switch you lose.

 

If you had a goat initially and switch you will win.

 

That is all that needs to be said about this problem.

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Guest LaBambaBar

In fact, let me break this down to its simplest form.

 

What does the question actually ask?

 

I flip two coins.  At least one of them lands heads.  What is the probability that both land heads?

 

OK - let's make it even more simple:

 

Two coins.  One of them IS heads.  What is the probability that both are heads?

 

Answer: 1 in 2.

 

End of.

 

Draw a line under it.

 

The fat lady has sung.

 

Elvis has left the building.

 

Stick a fork in this one.

 

Coz it.  Is.  Done.

 

Perhaps you missed this earlier, pdoggg!

 

(Emphasis on IS added for this post.)

 

:rolleye0012:

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Posted Today, 08:44 PM

Pdoggg, on 15 Jul 2015 - 3:21 PM, said:snapback.png

Looks like we agree on something.  :biggrin:

 

After the door with the goat is opened their is now 1 door left with a car and 1 door with left with a goat.

 

If you had the car initially and switch you lose.

 

If you had a goat initially and switch you will win.

 

That is all that needs to be said about this problem.

 

That's all that needs to be said if it is equally probable that you initially picked a car or a goat.

 

But that is not the case.

 

I think you will agree that there is a 1/3 chance that you initially picked a car and a 2/3 chance that you initially picked a goat.

 

You now appear to agree with

 

 

After the door with the goat is opened their is now 1 door left with a car and 1 door with left with a goat.

 

If you had the car initially and switch you lose.

 

If you had a goat initially and switch you will win.

 

So then by switching you have a 2/3 chance of winning.

 

But if you don't switch you will have only a 1/3 chance of winning.

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For the Monty Hall question imagine an exaggerated scenario where you have 100 doors and there were goats behind 99 and a car behind one. The theory is the same as the 3 door version.

 

You pick one door.

 

98 doors are then opened to reveal goats, leaving your original door plus one other.

 

Initially your door choice was made with 100 to choose from = 1/100

 

Sticking gives you a 1/100 chance

Switching gives you a 99/100 chance

 

- - -

The 3 door version:

Sticking gives you a 1/3 chance

Switching gives you a 2/3

 

Switching gives you better odds.

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Anecdotal 'evidence' we 'tried in the bar'?  I'm sure that'll be in line for next year's Nobel prize...

 

 

 

Well this is a third world tranny forum. So perhaps not as rigorous an exercise as you would prefer.   :biggrin:

 

Are you saying that that using Tomcat's wording that the probability of 2 heads is 50%?

 

Why do you object to simulation as a tool in arriving at the truth?

 

I do have some issues with using only 30 trials in the simulation.  With 30 trials there is about a 5% chance that it'd come up double heads 15 times or more.

 

However with 100 trials there is less than a one in a thousand chance of 50 or more double heads/

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No, my answer IS to the question posed by Lefty.  You and Tomcat are the ones who seem to have the wrong end of the stick.  In fact you seem to think the stick has 3 ends!

 

Anecdotal 'evidence' we 'tried in the bar'?  I'm sure that'll be in line for next year's Nobel prize...

 

The correct answer, as I have repeatedly shown, is a probability of 1 in 2, as I even explained in such a simple way that a feeble-minded 4-year-old could understand it.  Perhaps I was pitching it too high for a UKIP supporter.  555.

 

Nothing either of you can say will change the fact that I have answered correctly the question as given.

 

i wouldn't start insults as you dont even agree with the MH Problem and this is also not in doubt either.. your the idiot not me.. whats a being UKIP supporter have anything to do with this ???.. you have .lost the plot thats for sure.

 

Lefty question says that at least one of the coins is heads.. this means that the coins are in a supposition of three states

there are two coins remember.. this is the point of the question you are not told which coin is heads

 

CoinA    CoinB

T           H             state 1

H           T             state 2 

H           H            state 3

 

these are the only states allowed ... 

 

therefore HH is one in three

 

Its a probability question and nothing more .. Only the guy asking the question can peek and he just says " at least etc etc"

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I do have some issues with using only 30 trials in the simulation.  With 30 trials there is about a 5% chance that it'd come up double heads 15 times or more.

 

However with 100 trials there is less than a one in a thousand chance of 50 or more double heads/

 

I didn't pull the 5% and one in a thousand out of my :movethatass: 

 

The binomial statistical distribution fits this sort of coin flip problem perfectly.

 

The parameters are n and p with n being the number of trials and p the probability of success.

 

TC and I would use 0.333333 as p while R67 would use 0.5 as p if defining success as 2 heads.

 

My probability estimations above come from using the following formula:

 

91fa6c4d41b4a175298e9d4bd37976fe.png   Unless you've taken a probability course, you'll find the above notation confusing.    So I'd suggest you use the following website:   http://stattrek.com/online-calculator/binomial.aspx    
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I don't really want to get involved in this, but I will  :happy0148:

 

Taking the question literally gives odds of 1 in 2, 50/50, heads or tails, one flip of the remaining coin.

 

There is no remaining coin...both coins are flipped as the question states and at least one of the coins is a head.

 

 

. In a trial of 200 flips on the site below ( see sample) HH came up 55 and HT 88( where at least one of the pair was a head as the question asked).. probably a 1000 flips is better but i cant be bothered to pair them all off as i have a life to lead :biggrin:

 

 proving the point...the more you flip the more it will hit a third overall. See PDs post for the formula explaining this. Very handy indeed i must say !

 

.55 and 88 is not 50/50

 

https://www.random.org/coins/?num=100&cur=60-usd.0005c

 

I

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This link is for a 2 coin flip using GBP.

 

https://www.random.org/coins/?num=2&cur=60-gbp.1pound

 

If you do 100 iterations you'll get HH approx 25 times and a head and a tail about 50.

 

Around 25 times you'll see TT but that doesn't fit the condition of the question.

 

So you'll have 75 trials that fit the question's conditions with the probabilty off HH being 25/75 or 1/3.

 

You won't get this exact result but as the number of flips approaches infinity the closer you will get to 1/3.

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tx

 

you reminded me

 

in my example i paired the coins off in twos .. i forgot to mention that..,taking the top ten coins and working down in blocks of ten

 

also

( there is a another real baffler ,, the two envelope question ..but maybe we had better start a puzzle and corundum thread.......... dont google it yet as its a real baffler  ..it was posed in 1938 by A Mr Edmund Landau )  OMG my brain really got screwed on this one)

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Of two indistinguishable envelopes, each containing money, one contains twice as much as the other. The subject may pick one envelope and keep the money it contains. Having chosen an envelope at will, but before inspecting it, the subject gets the chance to take the other envelope instead. What is the optimal rational strategy for maximising the amount of money to be gained?

 

 

 

My response would be that no point in switching

 

But what about if you are allowed to take a peek in your chosen envelope and there is a cheque for 10,000 bucks.. do you switch then if allowed

 

( note; the answer is straightforward in the initial question.. or so it seems but there are some twists  which is why it is called " The Two Envelope Paradox" in some books ) ill elaborate later

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But what about if you are allowed to take a peek in your chosen envelope and there is a cheque for 10,000 bucks.. do you switch then if allowed

 

I don't think switching gives any advantage or disadvantage.

 

I don't think this is the same as saying their is a 50% chance of the unchosen envelope having 5,000 bucks and a 50% chance of the unchosen envelope having 20,000 bucks.

 

This isn't straightforward.  I'm getting a headache thinking about it.

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Common sense says that if you are offered a choice of two envelopes and one has twice the amount of the other then it makes no difference which one you choose and then allowed to swop if you feel like it afterwards.. the chances are evens

....

But the so called probability argument says that you should swop envelopes whether you are allowed a peek or not.

Lets say that the amount thats in your Envelope is X . . you will either make a bad swap and get 1/2 X$ or make a good swop and get 2X$ so on average you walk away with 1.25X... the expected gain is 25%.. this is why its called the Two Envelope Paradox

 

Is there a way out of this argument... this is the conundrum

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The Nazi salute thing has me thinking in reverse.....Thinking in reverse comes with aging....The older one becomes the better one becomes at thinking in reverse...........And I am getting pretty damn good at thinking in reverse.........

 

I had an off the wall friend growing up...One day he came to me with a new riddle...And his riddle was..."What is the difference between a duck?"

 

"Between a duck and what?" I asked.................

 

"Nothing," he answered.  "Just a duck."

 

"Impossible," I said.................

 

"No, it ain't," he answered once again.........

 

"So," I said in desperation, "I'll play your game...I give up...What is the difference between a duck?"

 

"The answer to my riddle is," he answered with a huge smile, "One of its legs are both the same."

 

I knew for certain then that something was seriously wrong with my friend who I dearly liked...And all these years later...while I have no real information regarding him...I suspect that he is either locked up somewhere to protect himself and the rest of civilization........or he is the president of Harvard.......

 

But reverse thinking has allowed  me to think about his riddle and his answer and now I think that I have finally figured out what his riddle means.........

 

In any event thanks to him and his riddle and to reverse thinking and to the Nazi salute I truly do know now, just how many balls of string it takes to reach the moon...........

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....

But the so called probability argument says that you should swop envelopes whether you are allowed a peek or not.

Lets say that the amount thats in your Envelope is X . . you will either make a bad swap and get 1/2 X$ or make a good swop and get 2X$ so on average you walk away with 1.25X... the expected gain is 25%..

 

Since this could be said about either envelope and obviously flawed, then there is a problem with the probability argument. 

 

We know there is some amount of money in each envelope.  Knowing the exact amount in one envelope really shouldn't change things.

 

I have to think a bit more exactly why the probability argument doesn't work.

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Since this could be said about either envelope and obviously flawed, then there is a problem with the probability argument. 

 

 

 

yes there is a flaw .i didnt read the answer until just now and it certainly caught me out.. damn problems  haha

 

But --- the paradox does hold under one condition but thats not likely to happen any time soon ( clue)

 

One expert on this problem is a Dr Michael Picard who is a Mathematical Logic expert at Vic. Uni in Canada and its his book i am reading now. He also has  a book out called " This is not a book" which i have not read

 

Lets hope no aspiring Bernie Madoff types are reading here..  :biggrin:

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that the outcome is either HT, TH, or HH.    We are told that TT did not happen.

 

So I see it as 3 equally probable possibilities so a 1 in 3 chance that both coins are heads.

 

R67 and I went around and around on this for hours with neither of us changing our mind. 

Mr PDogg is correct. You guys should know better than to go against him on things like this, chess, etc. He is one of the fartest smellers I know. 

 

Oops, smartest fellers.  :winking0023:

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