[JohnGalt2341]

I posted this in Players Talk the other day but I didn't get very many responses so I figured I would post it here as well.

Here's the scenario... Lets say there are 100 marbles in a duffle bag. This is what you know for sure. Some of them are Red and some of them are Blue. There are no other colors. Your job is to figure out how many Red ones there are and how many Blue ones there are. The rules are... without looking in the bag you have to reach in and pick out 1 marble at a time. You then write down the color of the marble and then put the marble back in the bag. Shake the bag up and repeat. How many times would you have to pick a marble out of the bag before you think you could guess the correct number of Red and Blue marbles in the bag... and you are 95% confident that you are within 5 of the actual answer? All guesses and or formulas are welcome.

Sorry if my wording of this question is terrible... if you have any questions let me know.

[subs]

u could only guess the % of red and blue not the actual number, right?

i may be wrong (quite likely) but wouldn't it depend on the results u were getting. i mean if u did it 100 times and got 1 you would be way more confident of being within 5 than if u had got 50...

to many superbowl props to think properly

[JohnGalt2341]

u could only guess the % of red and blue not the actual number, right? i may be wrong (quite likely) but wouldn't it depend on the results u were getting. i mean if u did it 100 times and got 1 you would be way more confident of being within 5 than if u had got 50... to many superbowl props to think properly

Since there are exactly 100 marbles in the bag the percentage of red and blue marbles and the number of Red and Blue marbles will be exactly the same. And yes... the results you are getting will make all the difference. For instance... lets say you pulled a marble out 25 times and every time you did it you pulled out a Blue marble. Many people might think the bag is filled with 100 blue marbles. However... there could be 8 red marbles(or more) in the bag. Now, if you pulled a marble out of the bag 1000 times and every single time it was blue you could be pretty confident that the bag was filled with AT LEAST 90 blue marbles. Make sense?

I love the Bill Hicks avatar by the way.

[buby74]

If it was an infinitely big bag or sampling with replacement I would use the beta distribution. As there are only 100 I would use the same approach of calculating the probability of observing x reds from n total balls assuming there is 100 reds then assuming there is 99 reds then 98 reds etc etc then normalise and work out the 95% credible interval. Off the top of my head the number to be 95% confident +- 5 could differ depending on the proporton but I haven't worked it out

[subs]

sorry buddy - i had a really stupid moment, somehow didn't see the 100 marbles bit...

too much staring at numbers today.

good problem, looks like buby has got a good answer.

like ur avatar too, has some interconnectedness connotations...

[Pherst]

I think the first issue is that you're talking possibility, not probability.
Probablility suggests that the preceeding events have an impact on subsequent events.
A good example of probability is Texas Hold'em Poker.
With this game, each card you see on the flop, turn, and river (plus what's in your hand) dramatically effects the probability or chance you will get the hand you are looking for. Hence, poker is seen as a game of skill in the sense players need to be able to do the maths on the spot to work out the percentage chance (or probability) of them receiving the desired result and thus betting accordingly/

Conversely, possibility is demonstrated in the game of roulette. Spinning 10 Reds in a row does not suggest a Black is any more likely to come up than Red in the subsequent spins. The preceeding spin has no impact on any future spins (if it did I wouldn't be sitting here typing this). This is what I would liken your bag of marbles to. If you take a marble out, note it's colour, replace it, shake the bag up, and redraw another, the result of this draw is independent from any of the others. The fact you may pull a blue marble several times in a row gives no indication of how many blues are in there as you're consistantly drawing from the same volume of marbles, not eliminating any elements like you can do in poker.

The only thing you can do is make a 'guesstimate' on the results shown, but you can never get to a result with any degree of certainty as no elements are being eliminated from your sample size. All you could ever say with any certainty is the colour you hold in your hand exists in the bag, thus giving you a ratio of at least 99:1 blue vs red or vice-versa.

Hope this helps.

[MonkeyF0cker]

I think the first issue is that you're talking possibility, not probability.
Probablility suggests that the preceeding events have an impact on subsequent events.
A good example of probability is Texas Hold'em Poker.
With this game, each card you see on the flop, turn, and river (plus what's in your hand) dramatically effects the probability or chance you will get the hand you are looking for. Hence, poker is seen as a game of skill in the sense players need to be able to do the maths on the spot to work out the percentage chance (or probability) of them receiving the desired result and thus betting accordingly/

Conversely, possibility is demonstrated in the game of roulette. Spinning 10 Reds in a row does not suggest a Black is any more likely to come up than Red in the subsequent spins. The preceeding spin has no impact on any future spins (if it did I wouldn't be sitting here typing this). This is what I would liken your bag of marbles to. If you take a marble out, note it's colour, replace it, shake the bag up, and redraw another, the result of this draw is independent from any of the others. The fact you may pull a blue marble several times in a row gives no indication of how many blues are in there as you're consistantly drawing from the same volume of marbles, not eliminating any elements like you can do in poker.

The only thing you can do is make a 'guesstimate' on the results shown, but you can never get to a result with any degree of certainty as no elements are being eliminated from your sample size. All you could ever say with any certainty is the colour you hold in your hand exists in the bag, thus giving you a ratio of at least 99:1 blue vs red or vice-versa.

Hope this helps.

Horrible.

[FourLengthsClear]

The problem here is that the method chosen to 'solve' the problem is completely inappropriate.

A process of random sampling and statistical inference is fine where the population is large. For a large population, a Z-test based (which is based on the standard deviation among x samples within that population) is reasonably reliable.

Where is the population is so small however, the samples needed to avoid a high probability of a very high standard deviation are excessively large. The whole point of direct inference is to avoid having to sample/count the entire population. Where that population is 100, such inference is counter productive (and unnecessary).

[bztips]

The fact that the population is 100 is pretty irrelevant.

This is a simple binomial. If we want to be 95% confident of being within +/-5%, then we set: 1.96*sqrt(p*(1-p)/n) = 0.05 and solve for n.

This gives: n = 1.96^2*p*(1-p) / 0.05^2.

To be conservative, we set p=0.5 so that p*(1-p) is at its maximum. Plugging this in yields n=384 if I multiplied correctly. So you need about 384 draws to be 95% confident that you're within +/- 5. Again, this may be a bit high if the true p is way different from 0.5.

[FourLengthsClear]

The fact that the population is 100 is pretty irrelevant.

This is a simple binomial. If we want to be 95% confident of being within +/-5%, then we set: 1.96*sqrt(p*(1-p)/n) = 0.05 and solve for n.

This gives: n = 1.96^2*p*(1-p) / 0.05^2.

To be conservative, we set p=0.5 so that p*(1-p) is at its maximum. Plugging this in yields n=384 if I multiplied correctly. So you need about 384 draws to be 95% confident that you're within +/- 5. Again, this may be a bit high if the true p is way different from 0.5.

So to establish with 95% confidence the make up of a population of 100, we must randomly sample is excess of 3 times that population? In line with my first sentence, it is a competely inappropriate method.

[bztips]

The method is inappropriate IF you have the choice to simply dump out the bag and count the marbles. But if that's not the case, if you're only allowed to blindly pull out one at a time, then it doesn't really matter how many marbles are in the bag, and it won't significantly affect the required sample size to get to the desired level of confidence.

[FourLengthsClear]

The method is inappropriate IF you have the choice to simply dump out the bag and count the marbles. But if that's not the case, if you're only allowed to blindly pull out one at a time, then it doesn't really matter how many marbles are in the bag, and it won't significantly affect the required sample size to get to the desired level of confidence.

What?
Consider two very different bags and see how little sense that statement makes considering the task is to establish the actual proportion.

Bag A has 4 marbles and the actual proportion of blues is 50%
Bag B has 100 million marbles and the actual proportion of blues is 0.05%

[bztips]

Well I thought it was obvious that I was excluding extremely small samples as well as extremely small proportions, but whatever.

[Pherst]

Horrible.

And your proposed solution to the question is where champ...???
Thanks for taking the time to post your single word response, or is this the extent of your vocabulary...???

[JohnGalt2341]

Conversely, possibility is demonstrated in the game of roulette. Spinning 10 Reds in a row does not suggest a Black is any more likely to come up than Red in the subsequent spins. The preceeding spin has no impact on any future spins (if it did I wouldn't be sitting here typing this). This is what I would liken your bag of marbles to. If you take a marble out, note it's colour, replace it, shake the bag up, and redraw another, the result of this draw is independent from any of the others. The fact you may pull a blue marble several times in a row gives no indication of how many blues are in there as you're consistantly drawing from the same volume of marbles, not eliminating any elements like you can do in poker.

Spinning 10 Reds in a row does not suggest anything only because you already know that there are an equal number of Red slots on the wheel as Black slots and streaks are going to happen. But what if you didn't know how many slots on the wheel were Red vs Black? What if you were playing on a homemade Roulette wheel and the number of Reds and the number of Blacks weren't the same? And worse yet... you couldn't look at the homemade wheel. You could only hear the results of the spins. Would 10 Reds in a row now not have any influence on you? What if after 100 spins it landed on Red 90 times? Would that have any influence on you? Would it mean that 90% of the homemade Wheel had Red slots? Possibly... but not necessarily. In my mind... if it landed on Red 90 times out of 100 on this homemade roulette wheel that I was not allowed to see it would strongly suggest that there are more Red slots on the wheel than black ones. To me... this is what this marble experiment is like. If I already know that the bag is filled with 50 Red marbles and 50 Blue marbles then the past results don't mean anything to me. However, if I don't know how many Blue and Red marbles are in the bag then drawing 90 Red marbles out of 100 almost certainly suggests that there are more Red marbles in the bag than Blue ones. Would 100 draws be enough to give a person an accurate guess of how many marbles are in the bag? Probably not. But I do think it would influence most people(myself included) on what they think they would draw next.

[bztips]

Would 100 draws be enough to give a person an accurate guess of how many marbles are in the bag? Probably not.

I already gave you the eff-ing answer. What is it about the number 384 that you don't understand?

[JohnGalt2341]

I already gave you the eff-ing answer. What is it about the number 384 that you don't understand?

Although I do believe your answer is correct(or very close to it)... I don't think everyone necessarily agrees with it.

[MonkeyF0cker]

And your proposed solution to the question is where champ...???
Thanks for taking the time to post your single word response, or is this the extent of your vocabulary...???

That's the extent of my vocabulary. I would "propose a solution." However, there are plenty of examples of bivariate sampling with replacement/Bernoulli trial/binomial (beta) distribution confidence/credible intervals (depending on whether you choose to use frequentist or Bayesian methodologies) in this forum and on the web already.

Don't worry about being incorrect or anything. At least, you were noble about it. LOL.

[Pherst]

That's the extent of my vocabulary. I would "propose a solution." However, there are plenty of examples of bivariate sampling with replacement/Bernoulli trial/binomial (beta) distribution confidence/credible intervals (depending on whether you choose to use frequentist or Bayesian methodologies) in this forum and on the web already.

Don't worry about being incorrect or anything. At least, you were noble about it. LOL.

Can't say I'm worried mate. I rationalised it in it's most basic form. So far you've contributed nothing. I'll leave you to play with your high school text books. Cya champ...

[MonkeyF0cker]

Can't say I'm worried mate. I rationalised it in it's most basic form. So far you've contributed nothing. I'll leave you to play with your high school text books. Cya champ...

You should ask someone if they would prefer to receive nothing at all or a steaming pile of shit...

I wonder what their answer will be.

P.S. - High school textbooks? Does that mean that you didn't graduate high school? Can't say that I'm surprised, mate.

[Pherst]

You should ask someone if they would prefer to receive nothing at all or a steaming pile of shit...

I wonder what their answer will be.

P.S. - High school textbooks? Does that mean that you didn't graduate high school? Can't say that I'm surprised, mate.

'Does that mean that you didn't graduate high school?' Erm... How did you arrive at that conclusion...??? One of your brilliant formulas??? What's your response going to be? Something brilliant no doubt.

Nearly 10,000 posts I see. Wow... Maybe the milestone you've been trying so hard to achieve can be done so with another witty piece of prose in this thread. Perhaps try turning your computer off and climbing the stairs out of your mother's basement (assuming you're able-bodied enough to do so). There's a whole wide world out there big boy!!! Go get 'em!!!

[MonkeyF0cker]

'Does that mean that you didn't graduate high school?' Erm... How did you arrive at that conclusion...??? One of your brilliant formulas??? What's your response going to be? Something brilliant no doubt.

Nearly 10,000 posts I see. Wow... Maybe the milestone you've been trying so hard to achieve can be done so with another witty piece of prose in this thread. Perhaps try turning your computer off and climbing the stairs out of your mother's basement (assuming you're able-bodied enough to do so). There's a whole wide world out there big boy!!! Go get 'em!!!

Oh, boy. Mother's basement jokes? Awesome.

All of this because of a one word response to your ignorant post.

How long are you gonna cry, "big boy?"

[antifoil]

say something about taxes then he will get you a answer with graphs and links to stuff.

otherwise he just trolls.

[MonkeyF0cker]

Resentment over calling out posts full of misinformation. LOL.

Should I fill them with words of encouragement next time for you, children?

[antifoil]

you know what guy. if you have a problem with spreading misinformation, you need to get a life. this county was founding on misinformation. it is the right of the people. who are you to try and stop it. you disgust me.

i get sick of people trying to push their liberal agenda with graph. you can take those graph back to europe where they belong.

[Pherst]

Your use of the comma amuses me for someone who clearly thinks they're superior to others. 17 more to go big boy!!! Guess I hit a nerve... Don't hate the player, hate the game...

[samserif]

Some thoughts:

1. The number of samples required depends only on the proportion and the size of the confidence interval, not the size of the population (as long as we're doing a simple random sample with replacement). In the original problem, the confidence interval is expressed in terms of the population size but it's not clear what's important: that it is exactly 5 marbles or that it's exactly 5% of the population. For example, if we tried to generalize to a bag of 10,000 marbles, are we still talking about estimating within 5 marbles or are talking about 5% of 10,000?

2. If we're talking about 5%, not 5 marbles, then the number of samples doesn't depend on the number of marbles in the bag. If my bag has 10,000 marbles, five of which are blue, then I can draw 73 red marbles in a row and proclaim that there are 0% blue marbles with a confidence interval between 0% and 5%.

3. Which leads to another point: as bztips notes, the worst case is a proportion of 0.5, which requires 384 samples to get a 95% confidence interval with a 5 marble radius. That's the upper bound. The lower bound is 73 (proportion of 0.0 or 1.0). The actual number of samples required lies between those numbers and the only way to know is to sample, calculate, sample, calculate, etc., until you're good to go.

4. If you think "There's no way I'm putting my hand in that damned bag 384 times", browse through this for methods that work better for smaller numbers of samples.

[antifoil]

maybe you can understand this.

[MonkeyF0cker]

LOL.

[Pherst]

LOL.

Think you just got owned mate. 16 to go...

[donjuan]

You really don't deserve anything other than the response you got from Monkey. Your post to the OP made my eyes bleed.

[Dark Horse]

Translated to sports betting, the question is about winning percentage in a field where sample sizes are small.

Use Z-score. Wins minus losses divided by square root of sample size.

Instead of fixating on a set number and percentage, think in the range of a minimum of 300-400 bets before you can trust whatever it is you're doing. That is assuming that the environment remains unchanged over the years, which in sports betting is often not the case.

[Pherst]

You really don't deserve anything other than the response you got from Monkey. Your post to the OP made my eyes bleed.

I assume you're talking to me...???