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Probability (split from free throws)


tar
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Just in attempt to understand probability better, what if you are a 50 percent shooter, but you only count every other shot. Are the odds exactly the same, however you select your 100 consecutive shots that count, or does alternating the ones that count allow for a greater probability that the misses and hits will alternate, and you might count a greater number of hits, or a greater number of misses in a row?

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Just in attempt to understand probability better, what if you are a 50 percent shooter, but you only count every other shot. Are the odds exactly the same, however you select your 100 consecutive shots that count, or does alternating the ones that count allow for a greater probability that the misses and hits will alternate, and you might count a greater number of hits, or a greater number of misses in a row?

 

The odds are exactly the same. You must see that for independent sequential actions, no one pattern is more likely than any other. What I mean is that if you toss a coin, each toss has a 50% probability of happening. If you keep tossing it, the outcome of any throw is totally independent of the result of the previous one, so there is no reason to think that there would be a alternate heads-tails sequence.

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DrKrettin,

 

I understand, and thought that was the answer, but I am a little confused as to the statement that any pattern is just as likely as any other. After all 100 hits or 100 misses in a row have the same probability, but other patterns, other combinations of shorter runs of hits and misses have to be more likely than 100 of just the one outcome. There must be for instance a greater likelihood of having a streak of 3 of one outcome and then a streak of two of the other outcome...and a further combination of alternating and short runs and an occasional run of more than 5 adding up to a hundred, than having a streak of 62 of the one outcome and then a streak of 38 of the other. In other words there must be an average run length of either outcome, and patterns that included several of this average run length, would be more likely than patterns that included rare run lengths.

 

Regards, TAR

for instance,

 

I just flipped a coin 100 times.

 

HTTHHHTTTTTHTHTHTTTTHTTHTHTTHHTHHTHTHHHTTHTHHTTTHTTTHTHHHHHTHHHTHTTTTHTHHTHHHTTTHTTTTHHHTHTHHHTTHTT

 

48 HEADS

52 TAILS

 

54 RUNS

 

29 RUNS OF 1

11 RUNS OF 2

9 RUNS OF 3

3 RUNS OF 4

2 RUNS OF 5

 

AVERAGE RUN AROUND 1.85

 

My expectations would be, that patterns that yielded similar stats, would be more likely than patterns that yielded widely divergent stats.

That is, from the one trial, I could guess that out of a 100 flips, there would be close to 50 heads and 50 tails, and maybe about 50 runs with the most numerous run being a run of 1, then the next numerous, runs of 2, the next numerous runs of 3, then 4 then 5. Runs of 6 would happen now and again, runs of 7 less likely and 8 even less likely, etc.

Edited by tar
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I understand, and thought that was the answer, but I am a little confused as to the statement that any pattern is just as likely as any other. After all 100 hits or 100 misses in a row have the same probability, but other patterns, other combinations of shorter runs of hits and misses have to be more likely than 100 of just the one outcome.

 

 

It might be easier if you just think of the coin toss — regardless of outcome — as an event with a 50% probability. Any specific string of n events will have the same probability of happening, because each event has the same probability. So HHTHTT has the same probability as HHHHHH or TTTTTT

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Any specific string of n events will have the same probability of happening, because each event has the same probability. So HHTHTT has the same probability as HHHHHH or TTTTTT

 

That's what I was trying to say, but your statement is clearer. It might help (or it might confuse) to consider another well-known issue of probabilities, the dealing of cards from a pack. Bridge players each get 13 cards from a pack of 52, and one of the most important features of their hand is the distribution of cards between the 4 different suits. The most likely is a 4-4-3-2 distribution, 21%, (not 4-3-3-3, 10%) but sometimes you get unusual distributions like 8-3-2-0. Occasionally there are reports of a 13-0-0-0 distribution, but always unsubstantiated. The point is that every one of the 635,013,559,600 possible hands of cards is equally likely, but you remember the unusual ones more clearly than the mundane ones, so you can convince yourself that there is some kind of pattern (e.g. the opponents always get better cards), even when there is none. I'm not sure that helps.

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My expectations would be, that patterns that yielded similar stats, would be more likely than patterns that yielded widely divergent stats.

That is, from the one trial, I could guess that out of a 100 flips, there would be close to 50 heads and 50 tails, and maybe about 50 runs with the most numerous run being a run of 1, then the next numerous, runs of 2, the next numerous runs of 3, then 4 then 5. Runs of 6 would happen now and again, runs of 7 less likely and 8 even less likely, etc.

 

Be careful with terminology when speaking about probability; expectation has a very precise definition and is not just what one would personally 'expect'.

 

It depends on what you're counting. Consider flipping a fair coin 3 times; these are all the possible outcomes:

 

HHH

HHT

HTH

HTT

THH

THT

TTH

TTT

 

Each particular outcome is equally likely at p = 1/8.Intuitively we think that HHH or TTT is more unlikely, but we see it's just as likely as HTH or any other sequence. But if you're counting the number outcomes with exactly 2 heads, say, then that outcome has p = 3/8 of happening (and 1-p = 5/8 of not happening). If we are counting the number of runs of 3 heads then there is only one outcome with p = 1/8.

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Thread,

 

So I get the basic idea, that any particular outcome of 100 tosses of the coin is exactly as likely as any other, but within the 100 tosses, there are strings of 2, strings of 4, strings of 5, strings of 10, strings of 50, strings of 25, and strings of 20. The laws of probability hold for each of these strings as well, and one can consider the odds of something happening or not happening in each string, broken up into any size and starting and ending anywhere within the 100 trial set. These various influences conspire to create certain "kinds" of patterns, that are more likely then others. As in bridge, where you wind up, most of the time with suit lengths of 3 and 4 with sometimes 5 and 2 and some rarer times singletons and voids. And often you can tell something about the distribution of suits in other peoples hands based on what you have in your own. If you have 6 hearts, there are only 7 hearts in the other three hands, and most likely someone has 3 and the other two have 2. It can break other ways, but it is more common to find something close to equal distribution of all things in all ways. If there are 40 high card points to be had, for instance, it will be likely that you have close to 10 points. If you have 13 points, you know you have more high cards than average, and you open the bidding. If your partner opens and you have less than 6 points, you know that on average the opponents have as many points as your team and you will have difficulty making a high contract, and pass if you have three of your partner's suit. In other words, any possible distribution is as likely as any other, but there is more than one type of distribution being considered, and everything still has to add up. Of the four kings, each must be somewhere, on each deal.

 

With the coin toss I did, there were more single outcome runs, a few less two in a row, a few less 3 in a row, a few less 4 in a row and a few less 5 in a row. With no runs of 6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33....

 

My guess is that there is something about the various probabilities that all must add up, that conspire together, to make 100 trial runs more likely consisting of internal runs of 1,2,3,4,5 and 6, than runs of 25, 50, and 5 runs of 10.

 

Regards TAR

Edited by tar
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Enough to consider the more possible patterns are as if a 100 trial run of coin toss could be expected to be like an approximately 50 character word, spelled with mostly As a few less Bs, a few less Cs a few less Ds and a few less Es. With maybe patterns that had an F and/or a G being less likely than patterns with no letters past I (9p H or T in a row.) In other words, words spelt with letters at the end of the alphabet, representing runs of 20 or more, would be less likely than patterns spelled with letters representing smaller runs.

DrKrettin,

 

But pi is neither random nor proven to be a normal number. I suppose a normal number would be what would be expected as a normal distribution of numbers and strings and combinations and such, as in a bell curve, normal distribution, where certain outcomes are more normal than others that occur in a rarer and rarer fashion, at the various deviations from the norm. If you pick 100 people at random, it is more likely you will get 68 within a standard deviation of 100 IQ points, than 68 with an IQ above 165.

 

Regards, TAR

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I don't know about that any other sequence thing. You cannot, for instance, expect that given enough time the first 12.1 trillion digits of PI will repeat backward as the second 12.1 with the exception of there being an 87654321 in every previous spot, where there used to be a 99999999 in the first 12.1 trillion digits. Such a sequence is way too long to expect there would be any reason for the thing to ever happen. Even given an infinite amount of digits to take PI to, your sequence is impossible to arrive at, as soon as the first digit after the 12.1 trillionith digit is not the one you were looking for.

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I don't know about that any other sequence thing. You cannot, for instance, expect that given enough time the first 12.1 trillion digits of PI will repeat backward as the second 12.1 with the exception of there being an 87654321 in every previous spot, where there used to be a 99999999 in the first 12.1 trillion digits. Such a sequence is way too long to expect there would be any reason for the thing to ever happen. Even given an infinite amount of digits to take PI to, your sequence is impossible to arrive at, as soon as the first digit after the 12.1 trillionith digit is not the one you were looking for.

"Infinite" is really big. So no. Your intuition is failing you here.

 

Any event that does not have a probability that is identically zero will happen with an infinite number of tries.

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swansont,

 

I understand the rule, and the idea, but in the case of PI the numbers are not random, they are coming up for a reason, that has to do with the ratio of the circumference of a circle to the diameter. The sequence I proposed has a probability of zero.

 

And although infinity is really big and if you can assign a probability to an event, it would, given an infinite amount of trials eventually, or on the next trial, achieve the low probability arrangement, there are finite limits to certain things in reality and reasons for a thing happening or not, that would make the probability actually zero, upon inspection.

 

For instance, no matter how many times you roll the dice, you will never get a queen of hearts. For that, you need a deck of cards.

 

Or the chances of a particular particle being anywhere in the next nanosecond, are one thing if considered mathematically, and another thing if considered using common sense and reality, where everything must fit together. It cannot "get" further than its velocity will take it, in the time period you are considering.

 

Or if you figure the chances of all the electrons on earth winding up in one country while all the protons wind up in another...you might be able to assign a probability, but intuition wise one would know that the integrity of the places would be destroyed if such an event were to occur, so, the probability is, for all intents and purposes, zero. Not mathematically, but when held up against reality.

 

In the original thread for instance, the odds of missing 100 in row, were contingent on the odds you assigned to the shooter, of getting one shot in. If a shooter never missed, up to that point when you assigned the odds, then the probability for her to miss 100 in a row, would be zero. But that would not be the reality of the situation. Someone could open the door to the gym and cause a distraction or a breeze that would affect a later shot.

 

So yes, I understand the principle, and the idea, that if you assign a probability, then the thing will eventually happen...but your probability assignment might be in error, and a condition you thought would be a constant might turn into a variable or become impossible due to other things happening in the universe over your infinite amount of trials. As in the OP of the original thread, the poster suggested that fatigue not be a factor. At one point I suggested that the gym would be torn down, before infinity could do its mathematical magic.

 

Regards, TAR

So my only point is that the numbers work, and one can "get" the reality that the numbers are after, but reality often defeats the numbers.

 

In my old job, Six Sigma goals for a certain amount of uptime for the computer system that ran our hotline put the numbers at 99.996 or something. I forget the exact number gotten to by a certain deviation, but one time the system was down for nearly 2 days, and at the time, I calculated the amount of continuous uptime that would be required to reach the goal, and it was on the order of thousands of years, like 20 or 30 thousand. I think that the probability of reaching the goal was, at that point, effectively zero. We would not be using that system for 30 thousand years, and the next failure would push the likelihood of reaching the goal, even further down the line.

 

In the shooting scenario, the foul shooter would likely get better given an infinite amount of practice, and the odds of missing 100 in a row would actually dwindle so that the original odds would be incorrect, and the odds would get longer and longer as you went along.

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Seems to be a simple confusion about whether we are just using a real life example to visualise a purely mathematical consideration (which was how i read the OP), or whether we are trying to actually model reality. Both interesting problems, but quite different.

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But there is the debate. It is not confusion. Do the numbers force reality, or does reality force the numbers?

If you are using reality to come up with the odds in the first place, I think it appropriate to continue to use reality throughout the stretch of trials. It is even within the realm of the discussion, to consider if infinity ever can actually happen. By definition, you can never "get" there, so if it takes an infinite amount of time for a thing to happen, it is not going to happen.

it "could", but its not going to

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And although infinity is really big and if you can assign a probability to an event, it would, given an infinite amount of trials eventually, or on the next trial, achieve the low probability arrangement, there are finite limits to certain things in reality and reasons for a thing happening or not, that would make the probability actually zero, upon inspection.

Do you have a link to this "in reality" proof? I am unfamiliar with it.

 

Talking about finite limits with infinite trials is nonsensical.

 

For instance, no matter how many times you roll the dice, you will never get a queen of hearts. For that, you need a deck of cards.

And then there's ridiculous.

 

Or the chances of a particular particle being anywhere in the next nanosecond, are one thing if considered mathematically, and another thing if considered using common sense and reality, where everything must fit together. It cannot "get" further than its velocity will take it, in the time period you are considering.

 

Or if you figure the chances of all the electrons on earth winding up in one country while all the protons wind up in another...you might be able to assign a probability, but intuition wise one would know that the integrity of the places would be destroyed if such an event were to occur, so, the probability is, for all intents and purposes, zero. Not mathematically, but when held up against reality.

Because you aren't doing an infinite number of samples.

 

In the original thread for instance, the odds of missing 100 in row, were contingent on the odds you assigned to the shooter, of getting one shot in. If a shooter never missed, up to that point when you assigned the odds, then the probability for her to miss 100 in a row, would be zero. But that would not be the reality of the situation. Someone could open the door to the gym and cause a distraction or a breeze that would affect a later shot.

Which are all moot. You are arguing about things external to the problem.

 

So yes, I understand the principle, and the idea, that if you assign a probability, then the thing will eventually happen...but your probability assignment might be in error, and a condition you thought would be a constant might turn into a variable or become impossible due to other things happening in the universe over your infinite amount of trials. As in the OP of the original thread, the poster suggested that fatigue not be a factor. At one point I suggested that the gym would be torn down, before infinity could do its mathematical magic.

 

Regards, TAR

So my only point is that the numbers work, and one can "get" the reality that the numbers are after, but reality often defeats the numbers.

 

In my old job, Six Sigma goals for a certain amount of uptime for the computer system that ran our hotline put the numbers at 99.996 or something. I forget the exact number gotten to by a certain deviation, but one time the system was down for nearly 2 days, and at the time, I calculated the amount of continuous uptime that would be required to reach the goal, and it was on the order of thousands of years, like 20 or 30 thousand. I think that the probability of reaching the goal was, at that point, effectively zero. We would not be using that system for 30 thousand years, and the next failure would push the likelihood of reaching the goal, even further down the line.

Which tells you you failed to achieve your goal, not that the math was wrong.

 

In the shooting scenario, the foul shooter would likely get better given an infinite amount of practice, and the odds of missing 100 in a row would actually dwindle so that the original odds would be incorrect, and the odds would get longer and longer as you went along.

Again, a moot point. Completely irrelevant to the discussion of the problem.

It is even within the realm of the discussion, to consider if infinity ever can actually happen.

 

No, actually, it isn't. Once someone has set the parameters of a problem, questioning or changing the parameters is not part of the discussion of the answer.

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Swansont,

 

I always retain the right to question the validity of a premise upon which the other is going.

 

Besides, I was using the foul shooting example within the scope of a general discussion on probability, that I was split into, I therefore cannot go off topic, if I am discussing probability in general. And in that, a discussion of the meaning of infinite trials, is certainly central to the idea that if a probability is not zero then it is one.

 

My intuition says that there is a difference in the probability of a thing happening if it happens 1 in a million or if it happens once in a googleplex. Saying that both events could happen on the next trial, might be mathematically a true statement, but in terms of expectations, I would bet my life on the googleplex thing not happening, whereas I would not be so bold with the one in a million.

 

someone suggested, that ANY sequence can happen, given infinity, and I think that is a thing that can happen only in one's mind

 

in reality, there are influences that favor certain sequences over others, and when all the influences work in concert, there are some sequences where it is just unreasonable to ever expect them to occur

 

like swansont being the sole winner of the jackpot in every lottery in the country for the next 150 years

 

you could assign a probability and therefore it could happen...but could it?

 

regards, TAR

too many things about reality and what normally happens, would have to change for it to happen

 

Therefore it will not happen, even if the odds are mathematically 1.

and the odds instantly become 0 the first time there is a drawing where you don't have a winning ticket

 

What happens the next 150 years does not even matter once the first loss makes the sequence impossible.

Dr. Krettin,

 

Yeah, I knew about those dice when I made the comment. They were not the set I was rolling.

 

Regards, TAR

Edited by tar
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Swansont,

 

I always retain the right to question the validity of a premise upon which the other is going.

 

Besides, I was using the foul shooting example within the scope of a general discussion on probability, that I was split into, I therefore cannot go off topic, if I am discussing probability in general. And in that, a discussion of the meaning of infinite trials, is certainly central to the idea that if a probability is not zero then it is one.

Did I say anything about going off-topic?

 

Anyway,"the idea that if a probability is not zero then it is one" is a flawed representation of the idea. The probability of an event is not the same as the expectation of the outcome of multiple trials of an event.

 

Further, the validity of a premise is a separate issue from the solution to the problem, given that premise. It's math. A connection to reality is not required.

 

 

My intuition says that there is a difference in the probability of a thing happening if it happens 1 in a million or if it happens once in a googleplex. Saying that both events could happen on the next trial, might be mathematically a true statement, but in terms of expectations, I would bet my life on the googleplex thing not happening, whereas I would not be so bold with the one in a million.

That's just basic probability. Nothing surprising in concluding that the event with the smaller probability has a smaller probability. It's a tautology. And having probability inform your betting habits is not anything radical.

 

 

someone suggested, that ANY sequence can happen, given infinity, and I think that is a thing that can happen only in one's mind

 

in reality, there are influences that favor certain sequences over others, and when all the influences work in concert, there are some sequences where it is just unreasonable to ever expect them to occur

You keep bringing up "in reality" and it's irrelevant when the question involves an infinite number of attempts. That's about as far from reality as you can get. You're basically ignoring the question in that case, and imposing a new set of boundary conditions.

 

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swansont,

 

Well sure the probability of a possible event happening is 1 in whatever trials. My argument is that reality is such that some things are not possible, because of the long odds. If you say it is possible, because you figured the odds and they are not zero, therefore it is possible, you have a tautology there. If when discussing probability the important thing is whether it is possible given infinite trials, or impossible given infinite trials, then the solution to the problem is given in the premises. As soon as you ascribe odds, and the odds are not 0 then you have started the problem, by saying it is possible, by ascribing odds.

 

The suggestion was made before, that any sequence could happen next. Do you think this is anywhere near the way reality works? Are there not types of sequences that are going to happen BECAUSE each flip is either a head or a tail?

 

What do you think of my one trial observation, that it is likely, when you flip a hundred times, that you will get 50ish runs of heads or tails and the length of the runs is going to be short, with the most runs at length of 1, then 2 then 3 then 4 then 5 with maybe on rare occasions runs of greater length? That this "type" of sequence, given 100 throws is more likely than sequences involving multiple runs of 31?

 

Regards, TAR

It is like considering that there are enough galaxies and enough planets in each galaxy to consider there must be another Earth somewhere, with exactly what is happening here, happening there. When in reality there are not enough galaxies, given a finite universe, to cover the odds.

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dimreepr,

 

A sperm whale popping into existence over the planet could not happen. You need the conditions found in the sea, and a mommy and daddy sperm whale, which you do not have over the planet. Plus whales start out as little whales. That is fiction and an example of an event with zero possibility of happening, even given an infinite amount of time.

 

 

I think the universe must be finite in size because we have no evidence of matter being able to move at an infinite speed, and no matter how fast inflation was, it could not have been infinite in speed, because such a speed has no meaning. Therefore, if the universe 13.8 billion years ago was the size of a speck, it could not have grown to an infinite size, in a finite period of time. Therefore, the observable universe must be a finite percentage of the whole universe. Even if the whole universe is 100,000,000 times the size of the observable universe, it is still finite in size and the odds of everything somewhere else, happen in the exact sequence of how they happened here, with every person with exactly the same name and the exact same arrangement of elements and isotopes is beyond the realm of possibility...the number you could assign to the odds of an exact Earth existing elsewherewould be smaller than odds can be and there are not enough galaxies to cover these odds.

 

Guaranteed.

 

Besides, there would be at least one thing that could never be the same. When I would point to the other identical Earth, my doppelganger would be ponting in the opposite direction, disallowing the exact happenings in both places.

 

Regards, TAR

 

regards, TAR

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dimreepr,

 

A sperm whale popping into existence over the planet could not happen. You need the conditions found in the sea, and a mommy and daddy sperm whale, which you do not have over the planet. Plus whales start out as little whales. That is fiction and an example of an event with zero possibility of happening, even given an infinite amount of time.

 

 

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swansont,

 

Well sure the probability of a possible event happening is 1 in whatever trials. My argument is that reality is such that some things are not possible, because of the long odds. If you say it is possible, because you figured the odds and they are not zero, therefore it is possible, you have a tautology there.

 

 

IOW, you have two different scenarios. A person asks about scenario A (probability in an infinite number of trials) and you answer with scenario B (probability in a finite number of trials).

 

That's just being bad at answering the question.

The suggestion was made before, that any sequence could happen next.

I think the suggestion was more specific than that. Any specific sequence was as likely as any other sequence, with the sequences being the same length. You are just as likely to see HHTTHT as TTTTTT

 

Do you think this is anywhere near the way reality works? Are there not types of sequences that are going to happen BECAUSE each flip is either a head or a tail?

I don't know what you mean by that. But one flip does not influence another flip. If you flip T, that does not change the probability of the next flip being either value.

 

What do you think of my one trial observation, that it is likely, when you flip a hundred times, that you will get 50ish runs of heads or tails and the length of the runs is going to be short, with the most runs at length of 1, then 2 then 3 then 4 then 5 with maybe on rare occasions runs of greater length? That this "type" of sequence, given 100 throws is more likely than sequences involving multiple runs of 31?

I think that's a pretty straightforward conclusion.

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