Brain Teaser No 3

I'm off to have a shower. I need to feel clean again....

quote:

I can't concentrate when I'm shuddering.

A common, but very honest response to this little teaser, Deane.

....I'll put you down for slighlty longer than 60 seconds, shall I?...........

Cheers

Don

3 seconds - maybe the bottle of wine has helped ....


Ian

Radius length... as long as it took to say it, rather than type it! Ten units for any doubters. Sorry from Fredrik

Puck question!

I have got it at last. No experiements, only some thought.

Three cups, one puck.

Your friend knows which cup has the puck but you don't.

For you there is a one third chance that any of the cups has the puck under it, so you choose one, but do not look. Your friend then picks one which he knows has not got the puck under it. So the position is not exactly as choosing between two randomly, as the first cup which you chose was from a choice of three, and your friend then removes one that is "definately no good" [for you not having the puck under it], so the odds are that the other cup has more chance in the order of two thirds, of containing the puck! It is twice as likely to benefit you as the original choice...

Is that right? It has really been getting to me over the last couple of days!

As per this illustration below:

P-O-O O-P-O O-O-P

Though this could be replicated three times I am going to take it that in each case you you choose the centre mug, and in the first case your friend knows that the he must chose the right one, in the second case he must then choose the right one OR the left one, and in the third case he must choose the left one.

So only in the second case did you choose the winner at the first go - what would be statistically likely as a one in three chance. In both other cases you would win by changing at the second chance. So to change is going to get you statistically twice the chance compared to sticking! Brilliant!!!

The crucial element is that you know your friend knows which two cups in each case do not have the puck under them, and chooses one of them in every case...

Kindest regards from Fredrik

Yep Fredrik you've got it now.

Ian

That's really refrigerated Fredrik! Surely the Pooopooop! must be the sound of a happy ship's whistle then!

It took me over 5 seconds and less than 10, but I was certain of the answer then. Almost answered "C"(as in "Find X. Here it is!").

A Truel

Ok, we all know that a truel is a three-sided duel.

Tarquin, Tsunami and F (all names are entirely fictitious, and any resemblance to real persons either living or dead is entirely intentional) have agreed to a pistol truel. After drawing lots to decide who shoots first, second and third they take their places at the corners of an equilateral triangle. They will fire single shots in turn in the same cyclic order until two of them are dead. The man whose turn it is to shoot may aim wherever he pleases. All three know that Tarquin never misses. Tsunami is 80 percent accurate whilst F is only 50 percent accurate.

Assume all three adopt their best strategy for survival, and that nobody is killed by a stray shot, who has the best chance of survival?

And for the more mathematically minded, what are the precise survival probabilities of each of our three members?

Cheers

Don

Best let F shoot third I think! F would certainly be dangerous with a gun!

F. [Smiley]

How dare you Fredrik. What are you trying to imply?

quote:

Best let F shoot third I think!

Names must come out of the hat. F has a 1/3 chance of shooting last. But he may shoot wherever he pleases. Fredrik, I should stand well clear..........

Cheers

Don

Well at least I have a clearer idea who F is, and in that case the likelyhood of random death decreeses. Never was a shooting man myself.

ATB from Fredrik

I did Mechanics and Pure myself, so I saw the Geometry straight off, but Stats provide me with uncharted problems! Hence my slowness with the Pucking problem!

quote:

Well at least I have a clearer idea who F is

I'll put you in the next one Fredrik, don't want old Tarquin taking you out, do we?

Cheers

Don

quote:

A Truel
Ok, we all know that a truel is a three-sided duel.

..............All three know that Tarquin never misses. Tsunami is 80 percent accurate whilst F is only 50 percent accurate.

Assume all three adopt their best strategy for survival, and that nobody is killed by a stray shot, who has the best chance of survival?

And for the more mathematically minded, what are the precise survival probabilities of each of our three members?

So, is it too difficult or too boring?......need a clue.....or just want the answer so that we can move on?

We know that Fredrik gave up on statistics before university...........now is the time to buy that book (Theory and Problems of Statistics by Murray Spiegel - McGraw-Hill) and put right that glaring, but exciting ommission in your education that could enhance your life-style, job opport.........sorry, got a bit carried away.

Cheers

Don

I'm pretty sure I know the answer but haven't had time to work out all the numbers yet. Don't tell us yet ....

Ian

Don wrote:

quote:

We know that Fredrik gave up on statistics before university...........now is the time to buy that book (Theory and Problems of Statistics by Murray Spiegel - McGraw-Hill) and put right that glaring, but exciting ommission in your education that could enhance your life-style, job opport.........sorry, got a bit carried away.

Dear Don,

I have to admit I would not know where to start with this! I was notorious for seeing straight through Mechanics problems, and I was content to leave it at that, but strangely in A-level taken in 2002 [as a mature student!] the syllabus did not cover standard deviation in Pure and Mechanics! [Note that even my solution to the Puck problem was done visually, as I can think like that for hours, or even over days].

Maybe one day I will post some pictures of Civil Engineering work I did at Uni [only for six months, as I completely failed to finance the bid properly]. The funny thing about the best piece of structural work I did was to construct a Balsa Model bridge. The design I made was the most efficient design ever produced at Coventry up to that time [2003]. I had designed the whole thing in my head, and merely applied arithmetic to optimising the design further in terms of dimmensions. After I had made the final model, I started the technical drawings!

You have given me a recomendation for a Stats book, so may I please ask for your recomendation for a book covering Standard Deviation etc!

Kindest regards from Fredrik

quote:

Don't tell us yet ....

So long as at least one person isn't bored, my lips are sealed.............but I admit that I found this is a bit tougher than the last one!

Cheers

Don

quote:

so may I please ask for your recomendation for a book covering Standard Deviation etc

Same book, Chapter 4 "The Standard Deviation and other Measures of Dispersion"

and Chapter 6 "Elementary Probability Theory" (to help with the Truel - not Standard Deviations)

But remember, life got a whole lot easier when Microsoft introduced their standard deviation function into Excel. So now we can concentrate on "why" we want this measure of dispersion, rather than "how" to calculate it!

Cheers

Don

In Engineering it is a question of Manufacturing Tollerances and Quality control, but in Stats I am sure it is crucial in alsorts of areas, especially human behaviour, and ability etc!!

Not so much my field, as the Engineering aspect!

Is the book expensive? On the other hand if it is not £50 it will have to come. I like tensing up the leetle grey cells a bit!

ATB from Fredrik

Fredrik,

I've just realised that the above book was written in 1961 and is probably out of print.

I'm sure that there are a lot of more recent books that are probably better written, although I admit I keep going back to Spiegel, mainly through familiarity.

Cheers

Don

quote:

You have given me a recomendation for a Stats book, so may I please ask for your recomendation for a book covering Standard Deviation etc!

One would hope the stats book would cover standard deviations...

The truel (triel?) is interesting. I'm pretty sure that the basic scenario is covered in an earlier thread, somewhere on this forum the quiz show thing is done to death.

I think the strategies are,

Ta : kill the highest risk, Ts then F.
Ts : shoot at the highest risk, Ta then F.
F : shoot the ground until one of the others is dead, killing Ts means you die on the next shot, killing Ta means you're likely to die on the next shot. Better not to bother...

Given that it will always be F's shot after one of the others is killed it would appear that his chances of overall survival are in the region of 50%. (They cannot be less than 50%, if he shoots at Ta and misses then he dies but if he shoots at Ts and misses then he may survive to have another go.)

So simple logic suggests that F is most likely to win since the other 50% is shared between Ta and Ts.

I hope that was clearer than mud?

Paul

Fredrik,

I studied Civil Engineering at Nottingham University, but took up flying - the sponsor sort of insisted!. The book was part of our recommended reading. All engineering students at Nottingham did a common 1st year course (electrical/ electronic/ mechanical/ chemical/ mining/ production etc) so I think everybody used this book.

Cheers

Don

Amazon lists a 'Schaum's Outline' version of the Murray Spiegel book, here which looks worth picking up from one of the resellers for a relative pittance. This stuff doesn't date.

Paul

quote:

'Schaum's Outline' version of the Murray Spiegel book

That's the one - well, virtually any one those listed would fit the bill. Mine is a 1961 version but the later versions, especially the hardback at 97p looks worth a punt.........

many thanks Paul

Cheers

Don