Probability
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11-03-2013, 07:22 AM
RE: Probability
(11-03-2013 02:27 AM)Heywood Jahblome Wrote:  
(11-03-2013 02:22 AM)Aspchizo Wrote:  Can you demonstrate that? or are you just going to assert your right?

The reason the claim is true is because as you draw out white marbles, there are less ways the bin can contain black ones(let black mean any other color here). For instance suppose the bin has only 3 marbles. Let W stand for white marble and B stand for a black one. There are only 4 possible combinations of marbles.

  1. BBB
  2. BBW
  3. BWW
  4. WWW
If each combination is equally likely, then each has a .25 probability of being the true combination. Please note that before starting the draws the probability of the bin containing just white marbles is .25. What happens if on the first draw we draw white? Well a white draw eliminates combination 1. With only three remaining possible combinations the probability that the bin only contains white marbles rises to .33. Drawing a second white marble eliminates combination 2 and the probability of the bin only containing white marbles rises to .5. Of course if you draw a third marble and it is white, then the probability the bin only contained white marbles is of course 1.

I hope this example shows that the Probabiltiy of X(all the marbles in the bin are white) is an increasing function of An(n white draws without ever observing a black one). This demonstration is in agreement with the mathematical proof I provided earlier.
Assuming an infinite number of black and white marbles, you would be more or less correct statistically in saying that the probability of attaining anyone of the 4 possibilities in the above scenario is 0.25. If, however, the number of marbles is finite, then the probabilities change each time you draw a marble.

Start off of with small numbers first. If you only have 4 marbles (2 black and 2 white), you have a 50/50 chance of your first draw of getting a white marble, but after that draw, the probabilities change.

Let's say you had those 4 marbles
Before draw 1, odds of a white marble = 50%
Draw one = White

Before draw 2, odds of a white marble = 33%
Draw three = Black

Before draw 3, odds of a white marble = 50%
Draw three = white

Before draw 4, odds of a white marble = 0%
Draw four = Black


Let me ask you one simple question. Are statistics and probabilities descriptive, predictive, or both?

“Science is simply common sense at its best, that is, rigidly accurate in observation, and merciless to fallacy in logic.”
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11-03-2013, 10:35 AM
RE: Probability
(11-03-2013 06:26 AM)Kreuzfel Wrote:  Starcrash, it's because of the no-memory thing that the 00 will have always a 1/38 probability of showing up regardless of what showed up before (in particular it does not become more likely to appear if it didn't appear before). For this very reason, the probability of not showing up is 1-1/38=37/38, the probability of not showing up for 2 times in a row is (37/38)^2 and so on... So the probability of not showing up after one million times is practically zero (1.34*10^(-11582))... So, if after a lot of plays it doesn't show up there is an high chance that we are dealing with a single 0 roulette (assuming the roulette is fair, of course).

My mistake. I didn't notice the difference between "00" and "0". Every time I read it in my head it came out as "zero".

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11-03-2013, 10:47 AM
RE: Probability
What you're describing here, BlowMe, is the difference between perception and reality.

You're saying "the probability of all white marbles is increasing" but what you SHOULD be saying is "the person drawing the marbles will think, or PERCEIVE, that it is more likely to be all white marbles".

Are the contents of your barrel of marbles changing after each marble is drawn (I don't mean that there is one less marble, but rather, that someone is adding or subtracting random marbles completely separately from the drawn marble)? I don't think that was your intent; I'm assuming you are positing a barrel of marbles of unknown quantity and color, but once we have said barrel, the contents remain fixed, unchanged, except for the fact that we're drawing marbles out for our experiment.

Given that assumption, then before we begin drawing any marbles, we can safely state that whatever is in the barrel, even though it's unknown, it's a stable, static quantity that won't change during our experiment (other than the changes made by drawing the marbles from the barrel).

So here is what's happening, including reality and including your perceptions (misperceptions):
Initial barrel with fixed quantity, unknown to the person drawing the marbles, is (by way of example) 1,000 marbles, 999 of which are white, one of which is black.

Start:
Perception: "Wow, I wonder what's in this barrel of marbles. Could be anything, really."
Reality: 999 white, 1 black.
Probability: 99.9% chance to draw a white marble. Exactly.

Draw 1 marble, it's white:
HJ: "Hmmm, well, now I know this barrel has some white marbles, but maybe there was only one white one. In fact, I really don't know what's in the barrel yet."
Reality: 998 white, 1 black.
Probability: 99.9% chance to draw a white marble. Rounded to two decimal places.

Draw 1 marble, it's white again:
Perception: "Look at that, another white marble. Maybe there's lots of white marbles in this barrel, or maybe just these two, but the fact that I got two whites in a row makes me think there are lots of white marbles."
Reality: 997 white, 1 black.
Probability: 99.9% chance to draw a white marble. Rounded to two decimal places.

Draw 1 marble, it's white:
Perception: "Three whites in a row. There must be lots of white marbles in this barrel.."
Reality: 996 white, 1 black.
Probability: 99.9% chance to draw a white marble. Rounded to two decimal places.

Draw 6 marbles, all white:
Perception: "Now that's 9 white marbles in a row. This can't be just coincidence, the barrel must be really loaded with white marbles. I wonder if there's anything else in there?"
Reality: 990 white, 1 black.
Probability: 99.9% chance to draw a white marble. Rounded to two decimal places.

Draw 90 marbles, all white:
Perception: "99 white marbles and nothing else. I'm really starting to think this barrel only has white marbles."
Reality: 900 white, 1 black.
Probability: 99.89% chance to draw a white marble. Rounded to two decimal places. Note that the probability of the next marble being white is going DOWN.

Draw 401 marbles, all white:
Perception: "That's 500 white marbles and nothing else. Now I'm certain there are only white marbles in here. I am really sure, almost guarantee, that the next marble will be white."
Reality: 499 white, 1 black.
Probability: 99.80% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 400 marbles, all white:
Perception: "That's 900 white marbles and nothing else. Yes, this barrel MUST only be white. I am sure, actually guarantee, that the next marble will be white."
Reality: 99 white, 1 black.
Probability: 99% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 50 marbles, all white:
Perception: "That's 950 white marbles and nothing else. I have never been more sure that the next marble will be white."
Reality: 49 white, 1 black.
Probability: 98% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 40 marbles, all white:
Perception: "That's 990 white marbles and nothing else. I have never been more certain that the next marble will be white."
Reality: 9 white, 1 black.
Probability: 90% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 5 marbles, all white:
Perception: "That's 995 white marbles and nothing else. I have never been more positively certain that the next marble will be white."
Reality: 4 white, 1 black.
Probability: 80% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 1 marbles, it's white:
Perception: "That's 995 white marbles and nothing else. I have never been more absolutely positively certain that the next marble will be white."
Reality: 3 white, 1 black.
Probability: 75% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 1 marbles, it's white:
Perception: "That's 995 white marbles and nothing else. I have never been more absolutely totally positively certain that the next marble will be white."
Reality: 2 white, 1 black.
Probability: 66.67% chance to draw a white marble. Rounded. Note that the probability of the next marble being white is going DOWN.

Draw 1 marbles, it's white:
Perception: "That's 995 white marbles and nothing else. I have never been more absolutely totally completely positively certain that the next marble will be white."
Reality: 1 white, 1 black.
Probability: 50% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is going DOWN.

Draw 1 marbles, it's white:
Perception: "That's 995 white marbles and nothing else. I have never been more absolutely totally completely positively certain beyoind any lingering doubt that the next marble will be white."
Reality: 0 white, 1 black.
Probability: 0% chance to draw a white marble. Exactly. Note that the probability of the next marble being white is now ZERO.
Sure, this is just one example, but it's simple and straight forward. I could do the same thing with any other number of marbles and any other number of varying colors, and the trend will be the same: the PERCEPTION of future white marbles keeps going up, but the PROBABILITY keeps going down.

BlowMe, that's what you have been saying all throughout this thread, that YOU PERCEIVE the probability going up but you're wrong - it's going down. Hopefully I've illustrated this point well enough that you will now see the error you've made, mistaking your perception for the actual probability.

I would have agreed with your initial post if you had written it like this: "What happens if you reach into the bin and pull out a white marble? You get information. You get information about the unknown composition of the bin. You know that the bin contained at least one white marble. It could contain marbles of other colors, or all the marbles that were/are in the bin could be white. You reach into the bin an pull out another white marble. You have more information about the initial composition of the bin....it had at least 2 white marbles and it seems to you that it is more likely that all marbles in the bin are white. In fact the more white marbles you draw without ever finding a non white marble, the more likely it seems to you that all marbles in the bin are white. " (those are your words from your initial post, with my edits).

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12-03-2013, 12:45 AM
RE: Probability
(11-03-2013 03:23 AM)Jakel Wrote:  There are only four possible outcomes yes. But the possiblity for the combinations is not the same.
There are 2^3=8 combinations yes:

WWW WWB WBB WBW BWW BWB BBB BBW

If each marbles drawn have the same possibility of being white or black, and the draws are independent
the probability of WWW is 1/8.

Another way to see it, is that if the probability of drawing white/black is 50/50 and independent, then the probability for WWW=0.5*0.5*0.5=0.125=1/8.

EDIT: I changed "combinations" to "outcomes" in the start

Ah, Now I see how you are arriving at 1/8. As far as I am concerned WBW and WWB are the same combination but I am not going to quibble about it. It doesn't change the fact that drawing a white marble without ever drawing a black one increases the chances that all the marbles are white.
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12-03-2013, 12:53 AM
RE: Probability
(11-03-2013 10:47 AM)Aseptic Skeptic Wrote:  What you're describing here, BlowMe, is the difference between perception and reality.

The reality is you don't know the composition of the bin until you draw all the marbles. Any statements you make about the composition are going to be probabilistic ones.

Suppose you and I observe a man begin to draw marbles one at a time out of a bin. We obervse him draw 17 white marbles in a row and not draw a marble of any other color. I then turn and say to you, "I'll bet you a c-note that all the marbles in the bin are white"

Would you make an even money bet?
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12-03-2013, 01:02 AM
RE: Probability
(12-03-2013 12:45 AM)Heywood Jahblome Wrote:  It doesn't change the fact that drawing a white marble without ever drawing a black one increases the chances that all the marbles are white.
Unless this bin has some undisclosed property about it, where the contents are dynamically changing throughout the process, drawing marbles isn't going to change the "chances" that all of the marbles are white. They either are, or are not. Your knowledge of the contents doesn't change the probability of drawing a certain colour. The only difference is that you can't ascertain the probabilities without knowledge of its contents.

If you have a bin full of white marbles, and you know that its full of white marbles, you will draw a white marble every time.
If you have a bin full of white marbles, and don't know that its full of white marbles, you will draw a white marble every time.
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12-03-2013, 01:03 AM
RE: Probability
(12-03-2013 12:45 AM)Heywood Jahblome Wrote:  
(11-03-2013 03:23 AM)Jakel Wrote:  There are only four possible outcomes yes. But the possiblity for the combinations is not the same.
There are 2^3=8 combinations yes:

WWW WWB WBB WBW BWW BWB BBB BBW

If each marbles drawn have the same possibility of being white or black, and the draws are independent
the probability of WWW is 1/8.

Another way to see it, is that if the probability of drawing white/black is 50/50 and independent, then the probability for WWW=0.5*0.5*0.5=0.125=1/8.

EDIT: I changed "combinations" to "outcomes" in the start

Ah, Now I see how you are arriving at 1/8. As far as I am concerned WBW and WWB are the same combination but I am not going to quibble about it. It doesn't change the fact that drawing a white marble without ever drawing a black one increases the chances that all the marbles are white.
That is simply not correct in this senario.

If you roll a die 3 times and get 6 6 6, you would not (I hope....) argue that the chance of getting a 6 in the next roll is larger than 1/6.
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12-03-2013, 01:04 AM
RE: Probability
(11-03-2013 07:22 AM)TheBeardedDude Wrote:  Assuming an infinite number of black and white marbles, you would be more or less correct statistically in saying that the probability of attaining anyone of the 4 possibilities in the above scenario is 0.25. If, however, the number of marbles is finite, then the probabilities change each time you draw a marble.

Start off of with small numbers first. If you only have 4 marbles (2 black and 2 white), you have a 50/50 chance of your first draw of getting a white marble, but after that draw, the probabilities change.

Let's say you had those 4 marbles
Before draw 1, odds of a white marble = 50%
Draw one = White

Before draw 2, odds of a white marble = 33%
Draw three = Black

Before draw 3, odds of a white marble = 50%
Draw three = white

Before draw 4, odds of a white marble = 0%
Draw four = Black


Let me ask you one simple question. Are statistics and probabilities descriptive, predictive, or both?

The problem we are trying to solve isn't what is the probability of drawing a white marble or a black one. We are trying to answer the question, does observing just white marbles being drawn(without ever observing a blackone) increase the probability that all the marbles are white?

I think that the mathematical proof I provided, along with the example I provided, proves conclusively that the answer to the question is yes.

I'm not sure why we are still debating this....its as silly as debating if evolution actually happens. If anyone wants to continue, show a fatal flaw in the last example I provided or the mathematical proof I provided.
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12-03-2013, 01:09 AM
RE: Probability
(12-03-2013 01:03 AM)Jakel Wrote:  
(12-03-2013 12:45 AM)Heywood Jahblome Wrote:  Ah, Now I see how you are arriving at 1/8. As far as I am concerned WBW and WWB are the same combination but I am not going to quibble about it. It doesn't change the fact that drawing a white marble without ever drawing a black one increases the chances that all the marbles are white.
That is simply not correct in this senario.

If you roll a die 3 times and get 6 6 6, you would not (I hope....) argue that the chance of getting a 6 in the next roll is larger than 1/6.

Your die example is flawed because you(presumably) know that the die is properly balanced and each side is labeled with different number that spans the counting numbers 1 thru 6. You in essence know the composition of the die. If you never examined the die and all you could ever see is the result side, then the more times a 6 comes up without any other number other than 6 coming up, the more likely that each and every side of the die is labeled 6.
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12-03-2013, 01:28 AM
RE: Probability
(12-03-2013 01:02 AM)WeAreTheCosmos Wrote:  Unless this bin has some undisclosed property about it, where the contents are dynamically changing throughout the process, drawing marbles isn't going to change the "chances" that all of the marbles are white. They either are, or are not. Your knowledge of the contents doesn't change the probability of drawing a certain colour. The only difference is that you can't ascertain the probabilities without knowledge of its contents.

If you have a bin full of white marbles, and you know that its full of white marbles, you will draw a white marble every time.
If you have a bin full of white marbles, and don't know that its full of white marbles, you will draw a white marble every time.

You are misunderstanding "probability" and "chances". Probability statements are based on the limited knowledge you have about a system and not complete knowledge of the system. If we had complete knowledge we would have no need for probability theory.
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