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 Tags Bernoulli Trial , degree of belief , probability , relative frequency

 29th July 2012, 07:10 PM #1 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 How do you define "Probability" without using a synonym? I know that in a Bernoulli Trial situation, probability can be defined as the relative frequency of successes if the trial is repeated indefinitely. But how do we define probability in one-off situations (for example, predicting that a certain event will take place at a certain date)? The concept of probability is still meaningful in these situations because if you can accurately calculate the probability of one-off events you can (in the long run) make money by betting on one-off events. Even though it may be possible to accurately calculate the probability of a one-off event, defining probability in this situation is still problematic. Many text books define probability as a "degree of belief" but that seems to me to just be a synonym for probability. Can anybody do better?
 29th July 2012, 07:31 PM #2 Halfcentaur Philosopher     Join Date: Jun 2010 Posts: 5,965 Originally Posted by psionl0 Can anybody do better? Not likely.
 29th July 2012, 07:41 PM #3 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by Halfcentaur Not likely. LOL
 29th July 2012, 07:50 PM #4 Richard Masters Illuminator     Join Date: Dec 2007 Posts: 3,030 Originally Posted by psionl0 I know that in a Bernoulli Trial situation, probability can be defined as the relative frequency of successes if the trial is repeated indefinitely. But how do we define probability in one-off situations (for example, predicting that a certain event will take place at a certain date)? The concept of probability is still meaningful in these situations because if you can accurately calculate the probability of one-off events you can (in the long run) make money by betting on one-off events. Even though it may be possible to accurately calculate the probability of a one-off event, defining probability in this situation is still problematic. Many text books define probability as a "degree of belief" but that seems to me to just be a synonym for probability. Can anybody do better? What do you mean by a one-off situation. Do you mean a situation where there is no pattern that has been established? I'm not clear on why probability is not an adequate word for this.
 29th July 2012, 08:02 PM #5 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 Many text books define probability as a "degree of belief" but that seems to me to just be a synonym for probability. I don't understand how, or why, you are differentiating between a synonym and and a definition. What is wrong with defining "probability" as a quantification of certainty about a proposition on a scale from 0 to 1, where 0 is certainty that the proposition is false and 1 is certainty that the proposition is true, and in between, the probability of a proposition is proportional to our certainty of it. It follows, for instance, that a probability of .25 means that we are three times as confident that the proposition is false as we are that it is true. Jay
 29th July 2012, 08:09 PM #6 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by jt512 What is wrong with defining "probability" as a quantification of certainty about a proposition on a scale from 0 to 1, where 0 is certainty that the proposition is false and 1 is certainty that the proposition is true, . . . That is certainly how we measure probability/certainty and many maths teachers define probability by the way it is measured. I would just like to be able to give my students more than a vague intuitive idea of what probability actually is.
 29th July 2012, 08:17 PM #7 Loss Leader Opinionated JerkModerator     Join Date: Jul 2006 Location: New York Posts: 11,885 Originally Posted by psionl0 But how do we define probability in one-off situations (for example, predicting that a certain event will take place at a certain date)? Aren't all situations one-off situations? I mean, the Red Sox and the Yankees are tied right now. One of them will win. Bookmakers are giving odds on the game. Their odds are based on the individual player histories, the franchise history, the manager's history ... but there's only one time in the history of the world the Yankees will play the Red Sox on July 29, 2012 and the winner of that event cannot yet be known. Nor can it ever be known how well the odds-makers predicted the outcome, or how likely a different outcome would have been. All we know is that on this one day, one set of events will happen. They're all one-offs in my opinion. ETA: And they're going to extra innings. __________________ Follow me on Twitter! @LossLeader This force is receiving all the right to vote through the use of magic. - Miernik Wieslaw VOTE FOR ME JUST BECAUSE Last edited by Loss Leader; 29th July 2012 at 08:20 PM.
 29th July 2012, 08:23 PM #8 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 That is certainly how we measure probability/certainty and many maths teachers define probability by the way it is measured. I don't understand that statement either. What do you mean by "measure" probability? What do we measure it with? I think you've got it backward. We measure (ie, quantify) our degree of certainty by probability, not the other way around. Probability is quantitative; it's the ruler. Certainty is thing needing measuring, like the length of a desk. Quote: I would just like to be able to give my students more than a vague intuitive idea of what probability actually is. Well, you could have them read Janynes, which derives the axioms of probability from the axioms of propositional logic. Short of that, I don't understand what is wrong with defining Bayesian probability as a quantification of (un)certainty. Frequentist probability, as you noted in your OP, can be defined as the long-run relative frequency of an event. Jay
 29th July 2012, 08:39 PM #9 Perpetual Student Illuminator     Join Date: Jul 2008 Location: USA Posts: 3,707 Here's another variation: Probability is a measure of the degree of confidence that we assign to an event's occurrence. We assign 1 for certain confidence and zero for no confidence and any number in between, which depends on the confidence level. __________________ It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ
 29th July 2012, 08:56 PM #10 JJM 777 Illuminator     Join Date: Jun 2004 Location: Finland Posts: 3,175 Originally Posted by psionl0 Can anybody do better? How about, companies which make money by successfully betting against consumers in the long run: - insurance companies - betting agencies
 29th July 2012, 09:53 PM #11 llwyd Critical Thinker   Join Date: Sep 2010 Location: Helsinki Posts: 259 I think the Bayesian probability as such is quite a simple consept: http://en.wikipedia.org/wiki/Bayesian_probability The problem arises as described in the article between objectivist and subjectivist approaches to it. I somehow feel that there is an "objective" probability to an event, if everything would be known, without it still being 1. But of course we can never know everything, so it seems that there has to always a certain subjective relationship to judging the odds. Having no mathematics this is just a common sense view, of course, and I would be interested to read comments by mathematically literate persons here...
 29th July 2012, 11:04 PM #12 DevilsAdvocate Illuminator     Join Date: Nov 2004 Posts: 3,651 Originally Posted by psionl0 I know that in a Bernoulli Trial situation, probability can be defined as the relative frequency of successes if the trial is repeated indefinitely. But how do we define probability in one-off situations (for example, predicting that a certain event will take place at a certain date)? The concept of probability is still meaningful in these situations because if you can accurately calculate the probability of one-off events you can (in the long run) make money by betting on one-off events. Even though it may be possible to accurately calculate the probability of a one-off event, defining probability in this situation is still problematic. Many text books define probability as a "degree of belief" but that seems to me to just be a synonym for probability. Can anybody do better? I guess you could define it as “largeness of opportunities”. Imagine a ball that falls from a random position above 4 slots. If the slots are evenly spaced, there are 4 options with equal “size”, so they have equal probability. If you double the width of slot 4, you still have 4 options but now slot 4 has twice the “size”, so it has twice the probability. Slot 4 has a greater “largeness of opportunities”. Similarly, if there are 5 slots of equal size numbered 1,2,3,4,4, there are still 4 options, but 4 has twice as many “opportunities”, so it has a has a greater “largeness of opportunities”. The word “opportune” is from the Latin meaning “favorable” from the phrase ob portum veniens “coming toward a port”. Each slot has an opportunity to get the ball, but some may be more opportune or favorable than others based on their size or frequency. Just as wind drives a ship into a harbor, probability drives a ball into a slot. The problem with this analogy is that it tends to anthropomorphize things. Is there some god who decides which options are the most favorable and who drives the ball to the most favorable slot most of the time? If not, then what does? What drives the ball? The word “probability” comes from the Latin “probare” meaning to test or try. This is similar to the definition you quote of “degree of belief”. The word “chance” comes from the Latin “cadens” meaning “to fall”. Like that ball I mentioned above. And in English has come to mean both “opportunity” and “randomness”. I think the puzzle you are facing is not the definition of “probability” but rather the definition of “randomness”. We can calculate the probability that an event can happen, but in a one-off test there can only be one result. What determines the result is “randomness”. In the real world, we don’t see randomness. People can learn to flip coins to get a certain result. Rolling dice is more difficult, but we could build dice rolling machines that get certain results fairly accurately. Even computers don’t generate truly random numbers. Perhaps in quantum physics there is something random in the real world that is truly random, but I don’t think we understand that yet. So the “randomness” that determines the result of a one-off event is the very “vague intuitive idea” that you are trying to avoid. I don’t think we can avoid it. But I also don’t think it is a difficult concept. __________________ Heaven forbid someone reads these words and claims to be adversely affected by them, thus ensuring a barrage of lawsuits filed under the guise of protecting the unknowing victims who were stupid enough to read this and believe it! - Kevin Trudeau
 29th July 2012, 11:07 PM #13 Halfcentaur Philosopher     Join Date: Jun 2010 Posts: 5,965 If I was explaining it to adolescents, I would tell them it's a way of guessing how likely or unlikely something was using actual available details that help me to form an educated guess.
 30th July 2012, 02:36 AM #14 TubbaBlubba Knave of the Dudes     Join Date: Jul 2010 Location: Communist Kingdom of Sweden Posts: 7,415 "Given the information available, the probability of something is our estimation that it will occur." I guess you could also say that it describes the relative frequency of an event in an infinity of multiverses where all of the information you know holds true. __________________ Disagreement begets progress.
 30th July 2012, 11:39 PM #15 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by various posters . . . likely . . . certainty . . . degree of confidence . . . largeness of opportunities . . . estimation . . . Hmmm, we are getting quite a few synonyms here. Originally Posted by some other posters . . . So what? . . . Why do you want to know? . . . unnecessary . . . I have noticed that the harder the question, the more posts of this nature that you see. Of course, if it was easy then even I could have answered it. I know how to teach the concept of probability to children. You run a number of scenarios by them (eg "rolling a 7 on a die") until they can correctly apply such words as possible/impossible, certain/uncertain, likely/unlikely or even chance to the scenarios. From there, it is a simple matter to introduce them to the probability scale. Unfortunately, a definition of probability will probably elude them - even if they become skilled statisticians.
 31st July 2012, 05:37 AM #16 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 Hmmm, we are getting quite a few synonyms here. As I explained in #29, in the Bayesian sense, probability is degree of belief scaled from 0 to 1. That is close to one definition you had (you also had a reasonable frequentist definition) all along. But for some incomprehensible reason that you have refused to explain, even when specifically asked, you have dismissed it as being a "synonym." You also made an incoherent objection (something about "measure") to my definition in #29, to which I objected, and you just ignored the objection and went on complaining that no one has given you a proper definition yet. The fact is, we have given you a definition. It is now your job to either accept it, or to explain coherently why it is inadequate. Just repeating "it's a synonym" or it's how we "measure probability" (whatever that means) isn't good enough. We've done what you've asked. The ball is now in your court. Quote: Unfortunately, a definition of probability will probably elude them - even if they become skilled statisticians. Don't worry. If they become skilled statisticians, they will learn the modern definition of probability based on the Kolmogorov axioms. Jay
 31st July 2012, 06:03 AM #17 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by jt512 As I explained in #29, in the Bayesian sense, probability is degree of belief scaled from 0 to 1. That is close to one definition you had (you also had a reasonable frequentist definition) all along. But for some incomprehensible reason that you have refused to explain, even when specifically asked, you have dismissed it as being a "synonym." You also made an incoherent objection (something about "measure") to my definition in #29, to which I objected, and you just ignored the objection and went on complaining that no one has given you a proper definition yet. It would appear that I inadvertently brushed you off. Sorry about that. I will deal with your previous post now. Originally Posted by jt512 I don't understand that statement either. What do you mean by "measure" probability? What do we measure it with? "quantify" is a better choice of words. Originally Posted by jt512 I think you've got it backward. We measure (ie, quantify) our degree of certainty by probability, not the other way around. Probability is quantitative; it's the ruler. Certainty is thing needing measuring, like the length of a desk. I would have thought that "probability" was like the word "charge" in which it could be the thing or the measure/quantity of the thing. I'm not disputing your use of the word but I have never heard it expressed that way before. I was just hoping that probability could be generally defined in purely mathematical terms just as it can be in the Bernoulli Trial case. I guess the best that can be done is to use some generalized formula.
 31st July 2012, 06:09 AM #18 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by DevilsAdvocate I think the puzzle you are facing is not the definition of “probability” but rather the definition of “randomness”. "randomness" is something else altogether. I'm not sure that there is anything such as a purely "random" force but it is not germane to the definition of "randomness". If the outcome of an event can neither be predicted nor controlled then it can be considered random. Many computer games make use of a pseudo-random number generator which is not random at all. However, since the average user knows nothing about the seed or algorithm of the generator, it is a truly random event for the user.
 31st July 2012, 07:17 AM #19 Beerina Sarcastic Conqueror of Notions     Join Date: Mar 2004 Location: A floating island above the clouds Posts: 23,835 Proper pseudo-random number algorithms pass tests that show their distributions don't significantly deviate from random expectations. However, they can be stretched too far easily if you don't know their limitations. I once did a simulation of some Dungeons and Dragons combat. This used the C rand() function built on one very well-known pseudo-random number generator. I forget the details, but for some hit percent, say 10%, the longer you ran it, the longer the chain of most hits in a row you would get. Fair enough. However, at some point, you got to 9 in a row, but never, ever got 10, no matter how long you waited, including many times longer than 10x what it takes to see 9 in a row. If your simulation relies on such strings, look for better generators. __________________ "Great innovations should not be forced [by way of] slender majorities." - Thomas Jefferson The government should nationalize it! Socialized, single-payer video game development and sales now! More, cheaper, better games, right? Right? Last edited by Beerina; 31st July 2012 at 07:29 AM.
 31st July 2012, 07:24 AM #20 Beerina Sarcastic Conqueror of Notions     Join Date: Mar 2004 Location: A floating island above the clouds Posts: 23,835 For the nerd: The internal value of the seed is 32 bit and the generator just returns the low (?) 16. The algorithm then generates the next based on that seed and returns the low 16 again. The seed for the next number is just the full 32 bit of what was returned to you on the previous call. When you call srand you're just initializing this value. Typically it generates numbers 55 at a time, so every 55th call to rand() will take some extra time. Anyway, this immediately suggests a random loop of 4 billion-ish then it repeats. Hence you would not be likely to see too many runs with a probability much lower than 1/4 billionish. If there were a lot of those, then the algorithm would fail random statistic checks, and would not have been used, of course. __________________ "Great innovations should not be forced [by way of] slender majorities." - Thomas Jefferson The government should nationalize it! Socialized, single-payer video game development and sales now! More, cheaper, better games, right? Right? Last edited by Beerina; 31st July 2012 at 07:33 AM.
 31st July 2012, 07:50 AM #21 Myriad HypertheticalModerator     Join Date: Nov 2006 Posts: 8,205 The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run. Respectfully, Myriad __________________ The cosmos is a vast Loom, with time the warp and space the weft. We are all fruit of the Loom, unaware.
 31st July 2012, 08:12 AM #22 Perpetual Student Illuminator     Join Date: Jul 2008 Location: USA Posts: 3,707 Quote: I was just hoping that probability could be generally defined in purely mathematical terms just as it can be in the Bernoulli Trial case. I guess the best that can be done is to use some generalized formula. If that's all you want, just call it: The ratio of the occurrence of a specific outcome to all possible outcomes. __________________ It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ
 31st July 2012, 08:15 AM #23 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 I would have thought that "probability" was like the word "charge" in which it could be the thing or the measure/quantity of the thing. I suppose that the word "probability" can be used in a colloquial sense to mean a qualitative likelihood or degree of certainty, but you were asking for a definition of "probability" in its mathematical sense. Our degree of confidence in a belief is not obviously quantitative. A rigorously justified quantification of subjective degree of confidence was not worked out until the mid-20th century (to the best of my knowledge). So "probability" in its mathematical sense is a quantification of a concept that isn't obviously quantitative, and furthermore, it is a specific quantification—a scale ranging from 0 to to 1. Quote: I was just hoping that probability could be generally defined in purely mathematical terms just as it can be in the Bernoulli Trial case. It can and it is. But the problem is that probability is an abstract concept, and according, abstract mathematics is needed to describe it.
 31st July 2012, 08:17 AM #24 Dave Rogers Bandaged ice that stampedes inexpensively through a scribbled morning waving necessary ankles     Join Date: Jan 2007 Location: In a world lit only by fire. Posts: 17,894 Originally Posted by Perpetual Student If that's all you want, just call it: The ratio of the occurrence of a specific outcome to all possible outcomes. The problem with that is the definition of the word "occurrence" as a variable that can take a continuous range of values, which is very different to the commonly used definition. In effect, you're using "occurrence" as, in part, a synonym for "probability". Dave __________________ "We will punish the murderer together. Our punishment will be more generosity, more tolerance and more democracy." - Fabian Stang, Mayor of Oslo SSKCAS, covert member
 31st July 2012, 08:28 AM #25 Perpetual Student Illuminator     Join Date: Jul 2008 Location: USA Posts: 3,707 Originally Posted by Dave Rogers The problem with that is the definition of the word "occurrence" as a variable that can take a continuous range of values, which is very different to the commonly used definition. In effect, you're using "occurrence" as, in part, a synonym for "probability". Dave Nonsense! The words occurrence and probability and quite apart from synonymous. There is no sense in which the word occurrence takes on a continuous range of values -- there is nothing preventing an interpretation of discrete values in my sentence. Addendum: For example, consider a die: The probability of rolling 3 is the ratio of the occurrence of 3 to the occurrence of all the numbers. __________________ It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ Last edited by Perpetual Student; 31st July 2012 at 08:41 AM.
 31st July 2012, 08:49 AM #26 Dave Rogers Bandaged ice that stampedes inexpensively through a scribbled morning waving necessary ankles     Join Date: Jan 2007 Location: In a world lit only by fire. Posts: 17,894 Originally Posted by Perpetual Student Nonsense! The words occurrence and probability and quite apart from synonymous. There is no sense in which the word occurrence takes on a continuous range of values -- there is nothing preventing an interpretation of discrete values in my sentence. Addendum: For example, consider a die: The probability of rolling 3 is the ratio of the occurrence of 3 to the occurrence of all the numbers. Discrete or continuous is not the point. The point is that the common definition of the word "occurrence" is "that which occurs", which does not allow any range of values; something either is, or is not, an occurrence, according to the common definition. Using that definition, your addendum would of course be nonsensical; so the definition of the word "occurrence" you've used is clearly more complicated than that. In fact, the definition of "occurrence" you've used in your addendum is "the number of times a certain outcome occurs in a given number of repetitions of the process", which - going back to the OP - is undefinable for a process that occurs only once. So, again going back to the OP, you can only define "occurrence" for a one-off event in terms of the probability of a certain outcome occurring, which means that the definition is circular. Hence, your definition of "occurrence", while not synonymous with "probability", is dependent upon its definition. Dave __________________ "We will punish the murderer together. Our punishment will be more generosity, more tolerance and more democracy." - Fabian Stang, Mayor of Oslo SSKCAS, covert member
 31st July 2012, 08:52 AM #27 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by Myriad The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run. Respectfully, Myriad I like this definition. "Expectation" is definitely not a synonym for probability. I think I will use this definition with my students.
 31st July 2012, 09:02 AM #28 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by Perpetual Student If that's all you want, just call it: The ratio of the occurrence of a specific outcome to all possible outcomes. Works for discrete cases but for definition purposes, you would have to assume that all the possible outcomes were "equally likely".
 31st July 2012, 09:05 AM #29 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 I like this definition. "Expectation" is definitely not a synonym for probability. I think I will use this definition with my students. Defining probability in terms of expectation is circular, since expectation is defined in terms of probability. What grade level are you students? What's their math background?
 31st July 2012, 09:15 AM #30 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by psionl0 Works for discrete cases but for definition purposes, you would have to assume that all the possible outcomes were "equally likely". You're right that his definition only works for the discrete case, but it does not depend on all possible outcomes being equally likely. Last edited by jt512; 31st July 2012 at 09:17 AM.
 31st July 2012, 05:37 PM #31 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by jt512 Defining probability in terms of expectation is circular, since expectation is defined in terms of probability. Obviously! However, it technically meets the requirements of the OP and defining both in terms of money adds an interesting dimension for students. Originally Posted by jt512 What grade level are you students? What's their math background? ATM I choose to deal only with senior high school students studying level 3 mathematics (the highest level). Not only is competition for such students low but ego is less of a problem for these students. I believe that I could also be successful with primary school children but I am less interested in making a game out of everything and there is much more competition for primary school students (teachers making money on the side etc.)
 31st July 2012, 05:40 PM #32 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by jt512 You're right that his definition only works for the discrete case, but it does not depend on all possible outcomes being equally likely. If you are defining probability merely in terms of counting outcomes, then all outcomes must be equally likely. Otherwise, the definition of probability becomes much more unwieldy.
 31st July 2012, 05:51 PM #33 Startz Critical Thinker   Join Date: Nov 2004 Location: Santa Barbara, CA Posts: 388 Originally Posted by Myriad The probability of an event is the maximum fraction of a dollar you should stake to win a dollar payoff if the event occurs, if you don't want to lose money in the long run. Respectfully, Myriad I think this is one of the accepted definitions. A slight modification, which avoids the issue of "long run," is The probability of an event is the fraction of a dollar you should stake to win a dollar payoff to make a fair bet. ("Fair bet" may be easier than "long run.")
 31st July 2012, 06:14 PM #34 cornsail Master Poster     Join Date: Jul 2009 Location: ohio Posts: 2,693 This thread may speak more to the difficulty of defining words in general than anything else.
 31st July 2012, 07:34 PM #35 jt512 Critical Thinker   Join Date: Sep 2011 Posts: 389 Originally Posted by Perpetual Student [J]ust call it [t]he ratio of the occurrence of a specific outcome to all possible outcomes. Originally Posted by psionl0 [Y]ou would have to assume that all the possible outcomes were "equally likely". Originally Posted by jt512 [i]t does not depend on all possible outcomes being equally likely. Originally Posted by psionl0 If you are defining probability merely in terms of counting outcomes, then all outcomes must be equally likely. Otherwise, the definition of probability becomes much more unwieldy. I believe you misinterpreted Perpetual Student's suggestion. He did not mean "counting outcomes," at least not in the sense you are using that ambiguous phrase. For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3. Jay
 31st July 2012, 07:46 PM #36 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by jt512 For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3. I think you just highlighted my point. (But I get it).
 31st July 2012, 07:49 PM #37 psionl0 Illuminator     Join Date: Sep 2010 Location: 31°58'S 115°57'E Posts: 4,793 Originally Posted by Startz ("Fair bet" may be easier than "long run.") I have found that when my kids say, "it's not fair!" I can usually shut them up by asking them what they mean by "fair".
 31st July 2012, 09:02 PM #38 Perpetual Student Illuminator     Join Date: Jul 2008 Location: USA Posts: 3,707 Originally Posted by jt512 I believe you misinterpreted Perpetual Student's suggestion. He did not mean "counting outcomes," at least not in the sense you are using that ambiguous phrase. For instance, his suggestion works for the following problem: A random experiment has two possible outcomes, A and B, that have either empirical or theoretical frequencies of x and 2x, respectively (for some non-negative x); then P(A)=1/3 and P(B)=2/3. Jay Exactly. If we consider the universe of possible outcomes of some event, it makes no difference if possible outcomes are equally likely or not. Simply take the ratio of the number of occurrences of interest to the universe of all outcomes. If the die were loaded, repeated experiments would give us our answer in any case for that particular die. __________________ It doesn't matter how beautiful your theory is, it doesn't matter how smart you are. If it doesn't agree with experiment, it's wrong. - Richard P. Feynman ξ
 1st August 2012, 10:21 PM #39 Finn McR Guest   Join Date: Mar 2009 Location: East Coast USA Posts: 285 My math ran out before formal set theory, but isn't that the way to go? The probability of an outcome "A" is the ratio of the ways outcome "A" can occur to the universe of all possible outcomes. In basic probability problems there are many simplifying assumptions. Take die rolling. Most statistics problems assume that the probability of any given number 1-6 coming up in the roll of a d6 is 1/6. That ignores what one would consider the die "coming up" or "showing." What if it is cocked against another die, the side of a box that the die is rolled in, character sheets, pencils, D&D manuals on the table, etc. Not to mention my friend's cat that would occasionally pounce on a die and and whack it into some dark corner. Bemoaning dismissiveness as an easy-out is a bit dismissive in itself. The probability of an event A that satisfies conditions of outcome "A" IS the ratio of the outcomes satisfying "A" to all possible outcomes.

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