1. ## Recreational Mathematics

I've tried to extricate this thread from the gambling on the Results of the Aussie Champs thread. It began (as most thread do) with a slinging match between Bill and Matt. But I'll start reproducing posts from the part where I got involved...

Originally Posted by Matthew Sweeney
Question:
A bag contains an unknown number of marbles >1.
Each marble is either black or white.
You draw out one marble, it is white and you put it in your pocket.
What colour is most likely to be pulled out in the next draw, and why?
Originally Posted by Barry Cox
The correct answer to this is that you have insufficent information to answer the question. All you know is that your chance of drawing a white marble has decreased and your chance of drawing a black marble has increased (provided the number of marbles in the bag is finite). But without knowing the initial mix and number of marbles you can't say either colour is more likely than another.

Assumption 1: The initial mix of black to white was 1:1

Answer 1: Black is now more likely as there are now 1 more black marble than white. However the change in probability is inversely proportional to the initial number of marbles in the bag.

Assumption 2: The initial mix of black to white was significantly different from 1:1 and there were a reasonably large number of marbles in the bag.

Answer 2: White was pulled out first so it is more likely that there were more white than black. Due to the large number of marble the non-replacement has not affected probablilities too greatly and therefore White is again the more likely result.

However, neither answer is correct unless the assumptions are accepted.

To illustrate the point for you here are some questions which have answers.

1. You have a bag with black and white marbles, 12 of one colour and 8 of the other. A marble is drawn at random (it happens to be white) and put into your pocket. Is it now more likely that the next marble drawn will be black or white? (4 points)

2. Provide the probability to 4 decimal places. (2 points)

Originally Posted by Matthew Sweeney
Almost true. We can always answer. I practical terms. the answer is "a white marble". However, your arguments and examples are 100% correct! I concur fully with you. We must make some assumptions.

I put it to you that practically, the bag it not of infinite size. Further that the proportion of white to black is unknown, and thus in practical terms normally distributed unless otherwise known. Therfore, of all the possible actual black and actual white marbles in the bag there are proportional very few that have B=W. This means that the number of balck and white are nearly always unequil. Now we can say, as you have already explained, that what ever draw out first is the best prediction for the second marble's colour.
Originally Posted by Barry Cox
Not so. We can answer or we can guess. When assuming you are making a guess as to what is reasonable. You cannot say what the probability of your guess being correct is. You may as well flip a coin.

Here you are making some additional assumptions as to the likely distribution of marbles. Furthermore, your logic is flawed. The reliability of the "same again" strategy is improved the more biased the sample of marbles and the greater the number of marbles, however, the first rule of Probability is do not trust your intuition.

BTW did you have a crack at my question? (hint)

Marbles are distributed 12/8, does the "same again" strategy perform better or worse than the "opposite" strategy?
There was an attempted solution submitted by Matt at this point but he clearly understood I was giving the distribution as Black:12, White:8 whereas I didn't intend this to be inferred from the way the questoin was worded. Anyway a second solution was submitted...

Originally Posted by Matthew Sweeney
#1. The most likely colour of the second marble is white.

#2. There are 4 possible outcomes for the two marble draw without replacement:
Majority marble, Majority marble
Majority marble, Minority marble
Minority marble, Majority marble
Minority marble, Minority marble

The probablity of these are :
12/20 x 11/19 = 0.3474
12/20 x 8/19 = 0.2526
8/20 x 12/19 = 0.2526
8/20 x 7/19 = 0.1474
SUM =1=P

The P of a Major coming out second is
(12/20 x 11/20) + (8/20 x 12/19) = 0.6000

The P of a Minor coming out is
1 - P(Maj) = 1 - 0.6000 = 0.4000

The first marble drawn was White,
and the P of drawing a Maj in the first draw is 12/20 = 0.6000,
hense the P of White being the Maj is also 0.6000.

It has been shown that that the porobabilty of drawing a Maj in the second draw is also 0.6. In deed, in the absence of information of what has already been draw, the P of drawing a Maj in any/every draw is always 0.6

Therefore the chance of the second marble being White is also 0.6000

Sorry Barry, your flipped coin and going against your instinct means the laser fitted sharks have just made yum cha of your girlfriend.
Originally Posted by Barry Cox

Q1 - 0 points - harsh but fair, the correct answer is Black

Q2 - 1 point - generous but some of your working is right, although your logic is confused.

1/6 is not a good start but don't lose heart, there will be future assignment with which you can earn credits prior to exam day.

The correct solution is as follows.

The colour of the marbles is largely irrelevant as we don't know if white or black is the majority. The easiest thing to do is to calculate the probablity of the second marble being the same colour as the first marble.

That is Maj then Maj, or Min then Min.

The probability (P1) is 12/20 * 11/19 + 8/20 * 7/19 = (132 + 56)/380 = 188/380

For completeness, possibilities for differing colours are Maj then Min or Min then Maj

The probability (P2) is 12/20 * 8/19 + 8/20 * 12/19 = (96 + 96)/380 = 192/380

NB: P1 + P2 = 1

NMB: P1 &lt; P2

So the most likely result (although only ever so slightly) is that second marble will differ in colour from the first. The first was White, therefore the second will most likely be Black.

Q1. It is more likely that the second marble will be Black

Q2. Black (P2) = 192/380 ~ 0.5053
White (P1) = 188/380 ~ 0.4947

Feel free to asks questions, but next time you're working on a probability problem, don't forget to check your intuition in at the cloak room as it usually leads one astray.

If you don't believe me do a web search on the "Monty Hall Problem". That one is a beauty.
Originally Posted by Kevin Bonham
I did this independently and got exactly the same answer as Barry, though I wouldn't have explained it as well as he did. Very neat example indeed.
Originally Posted by Barry Cox
Thanks, as an footnote, the inequality that is satisfied for differing colours to be more likely that same colours is quite neat. Here it is...

(x1 - x2) ^ 2 &lt; (x1 + x2)

where x1 and x2 are the numbers of marbles of each colour.

Or to express in words, the second marble is more likely to differ in colour from the first when the square of the difference in number of the marbles of each colour is less than the total number of marbles.

In this example x1 = 12, x2 = 8

RHS = 12 + 8 = 20 (total number of marbles)

LHS = (12 - 8 ) ^ 2 = 4 ^ 2 = 16 (square of the difference (in this case 4))

Anyway, LHS is less than RHS in this example so different marbles rule.

Kevin, if you haven't already you might like to derive this inequality to see if I have this right.
Originally Posted by Matthew Sweeney
That is a really tough marking system :shock: . I would have though maybe 3/6 . 2 for using a matrix, 2 for calculating their probabilities, -2 for logic error, 0 for answer, bonus 1 for showing that all draws have exactly the same chance of a Maj being picked.

Yes, I can see how I made it hard for myself, and picked the wrong matrix pair. I like how your more eligant approach did not lead to the logical complexities that I stuffed up. That Monty Hall Problem is a cracker. You bloody professional mathamaticians come up with fabulous ways to fool people.

However, you still have yet to say why you picked Black as the second colour in the absence of sufficient information.

You could reasonably assume the ratio of Maj to Min is normaly distributed. Futhermore, you might assume that estimates of the number of marbles on the bag would have an asymetric distribution. Putting those equations and your personbal estimate of the number of marbles, together with the inequality you gave {(x1 - x2) ^ 2 &lt; (x1 + x2)} you might have found the solution, while your girlfriend was dangling - but I doubt it. So why did you pick black.

2. Originally Posted by Matthew Sweeney
That is a really tough marking system :shock: . I would have though maybe 3/6 . 2 for using a matrix, 2 for calculating their probabilities, -2 for logic error, 0 for answer, bonus 1 for showing that all draws have exactly the same chance of a Maj being picked.
I think I said "Harsh but fair". 4 points for a true/false question which you have a 50% chance of guess. The calculation of the exact probability was 2 marks and you scored 50% in that department.

The result regarding a maj marble being drawn on the second time is intersting but was not asked.

Originally Posted by Matthew Sweeney
Yes, I can see how I made it hard for myself, and picked the wrong matrix pair. I like how your more eligant approach did not lead to the logical complexities that I stuffed up. That Monty Hall Problem is a cracker. You bloody professional mathamaticians come up with fabulous ways to fool people.
It is an interesting problem because at the heart of the disagreements on the correct solution is really, "what can be reasonably assumed?"

Originally Posted by Matthew Sweeney
However, you still have yet to say why you picked Black as the second colour in the absence of sufficient information.
I didn't say I'd pick black. I said I would mentally toss a coin and go the opposite to my original guess. But really, if the number of marbles could be reasonably guessed at based on sensory cues and the the distribution could be assumed to be almost normal (within the contraint of the bag containing at least 1 marble of each colour), then a maximising strategy could theortically be worked out based on the colour of the first marble. My example shows however, that the strategy would need to be worked out and not guessed at as intuition leads you astray as often as not. Also, be aware that the Dr Evil is a sick dude and any distribution he concocts is unlikely to be "normal".

Originally Posted by Matthew Sweeney
You could reasonably assume the ratio of Maj to Min is normaly distributed. Futhermore, you might assume that estimates of the number of marbles on the bag would have an asymetric distribution. Putting those equations and your personbal estimate of the number of marbles, together with the inequality you gave {(x1 - x2) ^ 2 &lt; (x1 + x2)} you might have found the solution, while your girlfriend was dangling - but I doubt it. So why did you pick black.
You are self-contradictory here, if you exclude symetric distributions then you are not talking about anything like a normal distribution. Also you said the bag does contain at least 1 marble of each colour so again normal distribution does not strictly apply. But I have another problem for you. This one is worth 8 marks so if you get it 75% correct you can redeem yourself to a pass mark.

Q3

Dr Evil has a marble making machine which randomly produces a black or white marble with equal probability. He uses this machine to fill a bag with 20 marbles. If at the end of the process the bag contains only one colour of marble then the marbles are emptied into the crack of Mt Doom and he begins the process again.

A bag which has been prepared in this way is presented to you and you draw a marble from the bag, it is White.

(a)
What colour marble is most likely to be drawn from the bag next? (2 marks)

(b) What is the probability that the next marble drawn from the bag will be White? (Show answer as a fraction and approximate to 8 decimal places) (6 marks)

Have fun.

Anyone can submit a solution to me my Private Message. If I don't hear from Matt with a solution within one week, a solution and results for other posters (if any) will be posted.

3. Originally Posted by Matthew Sweeney
That is a really tough marking system :shock: . I would have though maybe 3/6 . 2 for using a matrix, 2 for calculating their probabilities, -2 for logic error, 0 for answer, bonus 1 for showing that all draws have exactly the same chance of a Maj being picked.
You really expect 1 point for something as obvious as that?

Looking forward to your solution to Q3 Matt, after your doubting-Thomas effort towards my (admittedly fading) maths ability on the other thread. I reckon I can get this one without too much trouble and I doubt that I'll be the only one.

4. Originally Posted by Barry Cox
I think I said "Harsh but fair". 4 points for a true/false question which you have a 50% chance of guess. The calculation of the exact probability was 2 marks and you scored 50% in that department.

The result regarding a maj marble being drawn on the second time is intersting but was not asked.
Hmmm. I will let it rest with this comment: How many exams have you done where T/F questions were worth 4 marks while the interpretation and calculation questions were worth 2 marks? [one raised eyebrow]

Originally Posted by Matthew Sweeney
However, you still have yet to say why you picked Black as the second colour in the absence of sufficient information.
I didn't say I'd pick black. I said I would mentally toss a coin and go the opposite to my original guess. But really, if the number of marbles could be reasonably guessed at based on sensory cues and the the distribution could be assumed to be almost normal (within the contraint of the bag containing at least 1 marble of each colour), then a maximising strategy could theortically be worked out based on the colour of the first marble.
yes
My example shows however, that the strategy would need to be worked out and not guessed at as intuition leads you astray as often as not. Also, be aware that the Dr Evil is a sick dude and any distribution he concocts is unlikely to be "normal".
Dr Evil would want the highest execusion rate. So, he would assume that most people would pick what I did - same colour. Therefore Dr Evil's best stratergy would be to give a bag with an even number of colours.

Originally Posted by Matthew Sweeney
You could reasonably assume the ratio of Maj to Min is normaly distributed. Futhermore, you might assume that estimates of the number of marbles on the bag would have an asymetric distribution. Putting those equations and your personbal estimate of the number of marbles, together with the inequality you gave {(x1 - x2) ^ 2 &lt; (x1 + x2)} you might have found the solution, while your girlfriend was dangling - but I doubt it. So why did you pick black.
You are self-contradictory here, if you exclude symetric distributions then you are not talking about anything like a normal distribution.
I should have explained further, that the asymetric distribution of the marble number guess is due to the fact that you cannot guess a -ve number of marbles. However. we could assume a distribution that approximates a normal.

Q3

Dr Evil has a marble making machine which randomly produces a black or white marble with equal probability. He uses this machine to fill a bag with 20 marbles. If at the end of the process the bag contains only one colour of marble then the marbles are emptied into the crack of Mt Doom and he begins the process again.

A bag which has been prepared in this way is presented to you and you draw a marble from the bag, it is White.

(a)
What colour marble is most likely to be drawn from the bag next? (2 marks)

(b) What is the probability that the next marble drawn from the bag will be White? (Show answer as a fraction and approximate to 8 decimal places) (6 marks)

Have fun.

Anyone can submit a solution to me my Private Message. If I don't hear from Matt with a solution within one week, a solution and results for other posters (if any) will be posted.
I'll have a think about it. and post later. But for now, I will guess
(a) second marble will be black (after the white first)
(b) 0.50000001

5. Originally Posted by Matthew Sweeney
Hmmm. I will let it rest with this comment: How many exams have you done where T/F questions were worth 4 marks while the interpretation and calculation questions were worth 2 marks? [one raised eyebrow]
You can't claim my marking to be draconian when you had my love interest dangling over a pool of enraged sea-bass.

Originally Posted by Matthew Sweeney
Dr Evil would want the highest execusion rate. So, he would assume that most people would pick what I did - same colour. Therefore Dr Evil's best stratergy would be to give a bag with an even number of colours.
Often in Bridge you have to make decisions of ways of playing a hand based on the likely distribution of a suit between two hands. It is somewhat akin to the marbles of two colours problem. Usually it is the wild distributions (say 4-1 or 5-0) that cause problems as they can cause the declarer to lose trump control or allow the opponents to quickly set up a number of tricks at no-trumps. However, sometimes the ONLY way to make a contract is to assume the distribution is wild, when it actually turns out that way, it makes all the points you've given away in undertricks in the past just melt away.

Anyway, Dr Evil's best strategy is probably to not know the distribution himself. That way it prevents him from telegraphing the distribution via mannerisms, etc. Good bridge players (a class of which I am not a member, BTW) are masters of studying mannerisms for just such information.

Originally Posted by Matthew Sweeney
I should have explained further, that the asymetric distribution of the marble number guess is due to the fact that you cannot guess a -ve number of marbles. However. we could assume a distribution that approximates a normal.
Does my marble making machine in Q3 sufficiently approximate a normal distribution for you?

Originally Posted by Matthew Sweeney
I'll have a think about it. and post later. But for now, I will guess
(a) second marble will be black (after the white first)
(b) 0.50000001
If you're still guessing my efforts to educate you in the 1st law of probabilty have failed.

Interestingly, you are going to the colour different strategy now after introducing the pseudo-normal distribution factor. I thought you intuited that this would reinforce the same again strategy.

Regarding part (b), don't forget to provide the answer in fractions. Part marks will be given for wrong answers with partially correct method so it would help to supply as much working as you need to prove your answer.

Also I note you're answer for (a) and (b) are contradicting. Your answer for (b) is supposed to be the probability of the next marble being White. You gave an answer of Black in (a) but a > 0.5 probability for White in (b).

You might also want to submit your answer by PM and I post a solution message after one week. That way anyone else who wants to respond would get the chance. I don't know how many people interested in recreational mathematics we're going to find. But the problems might have an entertainment value to some, so perhaps that would be best.

6. ## Is probability intuitive

There is probably some biological advantage in being able to predict our immediate futures. Part of what forms intuition is accumulated subconscious knowledge and possessing certain intellectual atributes enabling application of that knowledge.

7. ## Re: Is probability intuitive

Originally Posted by David_Richards
There is probably some biological advantage in being able to predict our immediate futures. Part of what forms intuition is accumulated subconscious knowledge and possessing certain intellectual atributes enabling application of that knowledge.
Yes but go into any leagues club any night of the week and you will see scores of people doing nothing but feed machines their hard earned lolly with a very short memory with regardintuiting their prospect of turning a profit.

What is the selective advantage of wishful thinking?

8. ## Re: Is probability intuitive

Originally Posted by Barry Cox

Yes but go into any leagues club any night of the week and you will see scores of people doing nothing but feed machines their hard earned lolly with a very short memory with regardintuiting their prospect of turning a profit.

What is the selective advantage of wishful thinking?
There is the psychology of random reinforcemnt. Scumbags use pokies, which use this psychology, to sucker the punters.

A great example of the poewer of random reinforcement is an experiment done with rats a few decades ago. (You cannot repeat it nowadays becuase of animal ethics legislation.) A rats placed in a watertank with no way out wil drown in about a day - they "appear" to just give up swimming. If a rat is rescued just before it drowns, it experiances a positive reinforcement to swim. If the rescued rat is later watertanked a second time, it will swim for many days until it fails physiologically, not psychologically.

From this, we might theorise,then, that there is a survival advantage for neural networks, to remember any fortuitous event and act on it, even if it is a sample of one.

Intuative probability is very much a product of this advantagous servival stratergy. It is only in "artificial" instances - ie. complex man made senarios - that hard nosed mathematical statistics can be used to show that intuition is correct when it is correct and incorrect when it is incorrect.

Take three chess palyers: a 100% tactical player; a 100% intuative player; and a 50/50 player. All other influences being eqil. which one will have the lowest rating?

9. ## Re: Is probability intuitive

Originally Posted by Matthew Sweeney
A great example of the poewer of random reinforcement is an experiment done with rats a few decades ago. (You cannot repeat it nowadays becuase of animal ethics legislation.)
Not in any conventional university, at least. Cox University has no Ethical Review Committee.

Originally Posted by Matthew Sweeney
A rats placed in a watertank with no way out wil drown in about a day - they "appear" to just give up swimming. If a rat is rescued just before it drowns, it experiances a positive reinforcement to swim. If the rescued rat is later watertanked a second time, it will swim for many days until it fails physiologically, not psychologically.
This is sick, even for CU. I remember hearing about a similar experiment involving feeding birds (perhaps pidgeons) where the pigeons developed all sort of perculiar behaviour as this was randomly reinforced with irregular feeding times.

Originally Posted by Matthew Sweeney
Intuative probability is very much a product of this advantagous servival stratergy. It is only in "artificial" instances - ie. complex man made senarios - that hard nosed mathematical statistics can be used to show that intuition is correct when it is correct and incorrect when it is incorrect.
If intuition can be shown to be unreliable in controlled conditions doesn't mean it is useless. Just not as accurate as mathematical analysis. However, several factors may make mathematical analysis inappropriate to certain situations.

Originally Posted by Matthew Sweeney
Take three chess palyers: a 100% tactical player; a 100% intuative player; and a 50/50 player. All other influences being eqil. which one will have the lowest rating?
The lowest rated player will be the 100% intuitive one. I don't know why it just feels right.

10. No solutions received so far. So the prize for first correct answer is still available.

Anyway, on the advise of a friend I've reworded the question slightly, it doesn't change my intended meaning, but it might make it a bit clearer. Here it is in full, changes italicised...

Q3

Dr Evil has a marble making machine which randomly produces a black or white marble with equal probability. He uses this machine to fill a bag with 20 marbles. If at the end of the process the bag contains only marbles of one colour then all the marbles are emptied into the crack of Mt Doom and he begins the process again.

A bag which has been prepared in this way is presented to you and you draw a marble at random from the bag, it happens to be White.

(a)
What colour marble is most likely to be drawn from the bag next? (2 marks)

(b) What is the probability that the next marble drawn from the bag will be White? (Show answer as a fraction and approximate to 8 decimal places) (6 marks)

11. Originally Posted by Barry Cox
No solutions received so far. So the prize for first correct answer is still available.
hi Barry
Good luck with finding posters who will have a try at this.
If I submitted answers I would have an expectation of 1 mark out of 8.
Can you describe my answers and how I got them?
starter

12. Originally Posted by starter
Good luck with finding posters who will have a try at this.
Hope is the last thing to go.

Originally Posted by starter
If I submitted answers I would have an expectation of 1 mark out of 8.
Can you describe my answers and how I got them?
Would the first step entail inventing a machine that makes marbles?

13. Originally Posted by Barry Cox
No solutions received so far. So the prize for first correct answer is still available.
OK I'll send something in ten minutes.

14. I believe I've got the method but I'm just crunching all the numbers. There are probably many shortcuts I am missing that are making this more painful than it needs to be.

15. Originally Posted by Matthew Sweeney
OK I'll send something in ten minutes.
Got it, thanks.

Originally Posted by Kevin Bonham
I believe I've got the method but I'm just crunching all the numbers. There are probably many shortcuts I am missing that are making this more painful than it needs to be.
I didn't say it was going to be easy.

In fact I bounced it off a friend of mine who has a BMath(Comp Sci) he got it wrong through misinterpreting the question, or selecting the wrong method, or a combination of the two. So it is not a dead easy problem.

I'm looking into a proof that says something about the the behaviour of P(n) for all n > 1. I have it at the conjecture phase so far. But I'm pretty confident the conjecture is right. Hopefully, I'll have it ready to publish with the solution.