Probability for HJ (You can un-block for this one)
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09-03-2013, 03:06 AM
Probability for HJ (You can un-block for this one)
Your thread is invalid, HJ. The problem is that you do not have a mathematical knowledge of probability. You are making the Gambler's Fallacy, and if you followed your line of reasoning on the Vegas slots you'd probably leave with a much lighter wallet.


You cannot make a statement about the probability of drawing an X colored marble from a jar without knowing what marbles are in the jar. If you don't know the contents of the jar, then you cannot make any statements about the probability of the next marble you draw having color X. This is fundamental to probability theory.

Given you know the contents of the jar, then you can easily make statements about probability. If a jar contains 500 white marbles, 400 black marbles, 99 red marbles, and one pink marble, you can say that drawing a marble of those colors is a 50%, 40%, 9.9%, and 0.1% chance occurrence, respectively.

In this case, however, it's important to note that drawing one marble does influence the probability for the next draw. This is because objects are being chosen from a set, and you are altering the set each time an object is chosen. As the chooser, unaware of the contents of the jar, you are aware that these probabilities are changing, but you cannot say how they are changing because you are unaware of the contents of the box.

Again, your choice does change the probability of what the next result will be, but you still cannot make a prediction based on it.



Let's look at a slightly different set. When you flip a coin, you know that there is a 50% chance that it will land on heads, and a 50% chance that it will land on tails. You cannot predict what the next flip will bring, but you can predict that about 50% of the results will be heads and 50% will be tails. But let's say that you didn't know that a coin had a 50% chance of landing on either side. Say instead of seeing the coin flip, you just got a 1 or a 0 on a computer screen, based on someone else's flipping the coin. You would not be able to predict the 50/50 split.

However, this trial differs from the marble trial in two significant ways: you can flip the coin indefinitely (unlike the finite number of choices like in the marble trial), giving you a potentially infinite data set. And second, the result of one coin flip does not influence the next result. Because of that you can use the Law of Large Numbers to derive the 50%/50% odds from the data set. You can also make meta-analyses, for instance if you flip the coin 100 times there is a 33% chance that there will be 7 heads in a row. This is a bit more complicated, especially when you move to more complicated systems than just a coin toss. It requires some maths. But with an understood stochastic system (like the coin toss) you can analyze the system and develop a model (though still, notably, not predict the next result).

With a system like the marble jar, you cannot do this without having knowledge of the set (that is, how many marbles there are, what colors are present, and in what proportion they occur).





As for miracles, atheists do not dismiss them on probabilistic grounds. They are dismissed on a case-by-case basis when they are found to have a naturalistic cause (for instance the "drinking statues" in India were found to be made of porous stone that absorbed milk; or the moving sun miracle in South America which was pretty obviously caused by stuffing several thousand people into a field in 90*F heat with no water or sanitation facilities; and of course in many cases it's just pious fraud). After seeing enough miracles disproven, atheists may dismiss miracles entirely. From a probabilistic standpoint, this is in fact a fallacy. However, from a realistic standpoint, there are millions of miracle claims and only so many hours in an average lifespan; thus it makes sense to test/research the major claims, then dismiss the rest. Thus, miracles are not completely disproven as a class, but they are largely discredited.

E 2 = (mc 2)2 + (pc )2
614C → 714N + e + ̅νe
2 K(s) + 2 H2O(l) → 2 KOH(aq) + H2 (g) + 196 kJ/mol
It works, bitches.
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09-03-2013, 03:16 AM
RE: Probability for HJ (You can un-block for this one)
Well said. Probability theory, stochastic theory and statistics are pretty much my favorite math subjects. Yes

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09-03-2013, 04:32 AM (This post was last modified: 09-03-2013 04:48 AM by Heywood Jahblome.)
RE: Probability for HJ (You can un-block for this one)
(09-03-2013 03:06 AM)Phaedrus Wrote:  Your thread is invalid, HJ. The problem is that you do not have a mathematical knowledge of probability. You are making the Gambler's Fallacy, and if you followed your line of reasoning on the Vegas slots you'd probably leave with a much lighter wallet.


You cannot make a statement about the probability of drawing an X colored marble from a jar without knowing what marbles are in the jar. If you don't know the contents of the jar, then you cannot make any statements about the probability of the next marble you draw having color X. This is fundamental to probability theory.

Given you know the contents of the jar, then you can easily make statements about probability. If a jar contains 500 white marbles, 400 black marbles, 99 red marbles, and one pink marble, you can say that drawing a marble of those colors is a 50%, 40%, 9.9%, and 0.1% chance occurrence, respectively.

In this case, however, it's important to note that drawing one marble does influence the probability for the next draw. This is because objects are being chosen from a set, and you are altering the set each time an object is chosen. As the chooser, unaware of the contents of the jar, you are aware that these probabilities are changing, but you cannot say how they are changing because you are unaware of the contents of the box.

Again, your choice does change the probability of what the next result will be, but you still cannot make a prediction based on it.



Let's look at a slightly different set. When you flip a coin, you know that there is a 50% chance that it will land on heads, and a 50% chance that it will land on tails. You cannot predict what the next flip will bring, but you can predict that about 50% of the results will be heads and 50% will be tails. But let's say that you didn't know that a coin had a 50% chance of landing on either side. Say instead of seeing the coin flip, you just got a 1 or a 0 on a computer screen, based on someone else's flipping the coin. You would not be able to predict the 50/50 split.

However, this trial differs from the marble trial in two significant ways: you can flip the coin indefinitely (unlike the finite number of choices like in the marble trial), giving you a potentially infinite data set. And second, the result of one coin flip does not influence the next result. Because of that you can use the Law of Large Numbers to derive the 50%/50% odds from the data set. You can also make meta-analyses, for instance if you flip the coin 100 times there is a 33% chance that there will be 7 heads in a row. This is a bit more complicated, especially when you move to more complicated systems than just a coin toss. It requires some maths. But with an understood stochastic system (like the coin toss) you can analyze the system and develop a model (though still, notably, not predict the next result).

With a system like the marble jar, you cannot do this without having knowledge of the set (that is, how many marbles there are, what colors are present, and in what proportion they occur).





As for miracles, atheists do not dismiss them on probabilistic grounds. They are dismissed on a case-by-case basis when they are found to have a naturalistic cause (for instance the "drinking statues" in India were found to be made of porous stone that absorbed milk; or the moving sun miracle in South America which was pretty obviously caused by stuffing several thousand people into a field in 90*F heat with no water or sanitation facilities; and of course in many cases it's just pious fraud). After seeing enough miracles disproven, atheists may dismiss miracles entirely. From a probabilistic standpoint, this is in fact a fallacy. However, from a realistic standpoint, there are millions of miracle claims and only so many hours in an average lifespan; thus it makes sense to test/research the major claims, then dismiss the rest. Thus, miracles are not completely disproven as a class, but they are largely discredited.



Phaedrus, I appreciate your attempt at having a meaningful conversation here. I will take you off my ignore list.

Your error is that you are conflating the probability of drawing an X colored marble from the jar with the probability of all the marbles in the jar being the same color. These are very different probability problems. Since you address the wrong one, your whole critique fails.

I can't know the probability of rolling a 6 unless I know how many sides the die has. Thats basically what you are saying and I agree. But lets say I have a die whose number of sides are unknown to me. Everytime I roll the die and it comes up with a result of 6 or less, it increases the probability that I am rolling a 6 sided die.

Let X = the probability the die is 6 sided.
Let An = n rolls where the result was always less than 7.
The probability of X given An is an increasing function of n.

The same is true of the probability of all marbles being white, that it is an increasing function of n where An is the observation of white marbles without ever observing other colored marbles.

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09-03-2013, 05:04 AM
RE: Probability for HJ (You can un-block for this one)
Your problem is that you give insufficient information to make any sort of probability judgement at all on the matter. You are essentially saying this:

Given a set (marbles)
- with a known quantity of items in that set (1000)
- with an unknown set of colors (# of colors = range between 1 and infinity)
- with an unknown proportion of colors

Where each trial consists of drawing one marble at random, and permanently removing it from the set (in other words the system is non-Markovian and has memory)

Can you determine the set of colors and their proportions based on the results of your trials?



The answer is no. Common sense would dictate that if you draw 500 white marbles in a row, all or most marbles are likely white. But mathematically you cannot make that determination.




It is clear from your other examples that you do not understand probability theory. I don't mean this as an insult; just a statement.

The roulette wheel example from your other thread is in no way mathematically equivalent to the marble example. Here is what your roulette example is, mathematically:



Given a random number generator (the roulette wheel)
- Where there are two possible sets of numbers the generator could draw from:
--Set 1. [0, 1 - 37]
--Set 2. [00, 0, 1-37]
- Where the results of the last trial do not affect the results of the next trial (the system is Markovian and memoryless)

Can you determine which set of numbers is being used, based on the results of the trials?



The answer is yes, because you only have two potential sets, instead of an infinite number of potential sets. The system is also Markovian and you have an infinite number of trials to work with. So you simply run ten million trials and see if any 00s appeared. In this case, the longer you go without getting a 00, the higher the probability that the system is using Set 1. This is because of the properties of the test; it is not a general case.




Your die example is closer to the roulette wheel than the marbles since it is memoryless and you have infinite trials, but has some ambiguity. You do not eliminate the possibility of dice with multiple sides having the same number. You also do not elaborate on whether the dice are allowed to have up to infinity sides or whether we are drawing from a predefined list of sets (4, 6, 8, 10, 12, 20, 24 sided dice only, etc.). These variables influence what you can or can't say.





It is clear that you do not understand probability theory or stochasticity. You are currently unprepared for a debate on the subject. I recommend doing some research.

E 2 = (mc 2)2 + (pc )2
614C → 714N + e + ̅νe
2 K(s) + 2 H2O(l) → 2 KOH(aq) + H2 (g) + 196 kJ/mol
It works, bitches.
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09-03-2013, 05:28 AM (This post was last modified: 09-03-2013 06:01 AM by Heywood Jahblome.)
RE: Probability for HJ (You can un-block for this one)
(09-03-2013 05:04 AM)Phaedrus Wrote:  Can you determine the set of colors and their proportions based on the results of your trials?



The answer is no. Common sense would dictate that if you draw 500 white marbles in a row, all or most marbles are likely white.



I am not trying to show that you can determine a set of colors and their proportion based on the result of trials. I am trying to show that observing only white marbles increases the chances that all the marbles are white. I am trying to show what you call common sense is reality.

Let X = the probability that all the marbles are white.
Let An = n observations of white marbles without ever observing a non white marble.

Assume that for all n, P(X | An) > 0 and P(An+1 | An) < 1.

we then have

P(X & An+1 | An) = P(X | An)P(An+1 | X & An)= P(X | An).

we also have

P(X & An+1 | An) = P(An+1 | An)P(X | An+1 & An)= P(An+1 | An)P(X | An+1).

Combining these give us

P(X | An) = P(An+1 | An)P(X | An+1) < P(X | An+1).

Which is another way of saying, P(X | An) is an increasing function of n.

In the interest of honesty this proof comes from an instructor. I didn't write it myself.

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09-03-2013, 05:31 PM
RE: Probability for HJ (You can un-block for this one)
I admire mathematicians and logicians.
Recall a bit of 'probability' as part of logic course when ons student asked the tutor.....
"do you want to bet on it"? Happy debating gentlemen; you have inspired me to take another
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09-03-2013, 06:13 PM (This post was last modified: 09-03-2013 06:38 PM by fstratzero.)
RE: Probability for HJ (You can un-block for this one)
2:46

2 minutes 46 seconds in, it presents the argument that HJ is trying to present but failing at.

Actually what the problem of induction tells us is that we do not have absolute knowledge of the world, therefore all knowledge that use the method of induction cannot be absolute. We must always allow for our knowledge to be modified in case something in the future doesn't match our current understanding. Because the past may not match the future we should always leave our theories open to account for any new behavior or problems.

It's like having a computer program. The programmer can say he has absolute knowledge because he has the source code.

Suppose you wanted to try and predict what will happen. We'll you'd have to observe it, and describe it as close as possible, but you would have to leave your description open just in case the program does something unexpected. That way you can adjust your description to be more accurate to the original program. That's basically how science works.

Now it's not an either or. It's not either we know everything are we are wrong. It's more like we know this will happen 99.9999% of the time, but we won't claim absolute knowledge in case something that describes this better comes along. We might edit our work or throw it away for work that better describes that thing.

So it's not really a problem. If the future wasn't like the past, that is to say completely random, or seem to act on somebodies elses caprice we'd say that science is futile. Yet the truth is reality acts in patterns, changes in patterns, and we can measure those changes.




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The atheist is a man who destroys the imaginary things which afflict the human race, and so leads men back to nature, to experience and to reason.
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10-03-2013, 05:16 AM
RE: Probability for HJ (You can un-block for this one)
(09-03-2013 05:28 AM)Heywood Jahblome Wrote:  
(09-03-2013 05:04 AM)Phaedrus Wrote:  Can you determine the set of colors and their proportions based on the results of your trials?



The answer is no. Common sense would dictate that if you draw 500 white marbles in a row, all or most marbles are likely white.



I am not trying to show that you can determine a set of colors and their proportion based on the result of trials. I am trying to show that observing only white marbles increases the chances that all the marbles are white. I am trying to show what you call common sense is reality.

Let X = the probability that all the marbles are white.
Let An = n observations of white marbles without ever observing a non white marble.

Assume that for all n, P(X | An) > 0 and P(An+1 | An) < 1.

we then have

P(X & An+1 | An) = P(X | An)P(An+1 | X & An)= P(X | An).

we also have

P(X & An+1 | An) = P(An+1 | An)P(X | An+1 & An)= P(An+1 | An)P(X | An+1).

Combining these give us

P(X | An) = P(An+1 | An)P(X | An+1) < P(X | An+1).

Which is another way of saying, P(X | An) is an increasing function of n.

In the interest of honesty this proof comes from an instructor. I didn't write it myself.
I don't understand your notation. If X is the probability that all the marbles are white, then
P(X | An)=the conditional probability of the probability that all the marbles are white, given that the first n marbeles are white.
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10-03-2013, 05:49 AM
RE: Probability for HJ (You can un-block for this one)
Is this thread about the odds of getting a head job?
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