Probability (split from free throws)

[tar]

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.

 

 

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

 

 

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.

 

 

I think that's a pretty straightforward conclusion.

The odds were given that the event should happen once every finite number of trials. The infinite part was just the idea that any odds that were not zero, would be sufficient to ensure that eventually the happening would happen. It was the same question in both cases...would it happen?

 

I understand that one flip does not influence the next, except given a set of one hundred flips to study, one finds that alternation happen at close to the same rate as flipping the same thing. This likelihood that there will be about 50 heads and 50 tails in the sequence, seems to favor this class of sequences over the class of sequences that do not satisfy the likely distribution.

 

Regards, TAR

[Prometheus]

I understand that one flip does not influence the next, except given a set of one hundred flips to study, one finds that alternation happen at close to the same rate as flipping the same thing. This likelihood that there will be about 50 heads and 50 tails in the sequence, seems to favor this class of sequences over the class of sequences that do not satisfy the likely distribution.

 

There is no except. If the flips are independent then they do not influence each other at all - the sequence does not try to 'correct' itself to conform to the expectation. That is the gamblers fallacy.

 

I think you are confusing this for the law of large numbers (a very common thing, it's not really intuitive).

[tar]

Prometheus,

So I probably read about that a long time ago and am considering that in my question. If there is a trial of a million, the probability of any particular sequence is just as likely as 1,000,000 heads, but the law of big numbers will not allow the 1,000,000 heads. Somewhere along the line, a tail is going to show. That particular sequence of 1,000,000 heads is working contrary to the law of large numbers, and therefore will not happen. Could happen, but will not, because the 50 50 nature of the flip is too strong to allow it. Therefore, some other particular sequence, that has the same mathematical probability of occurring, but that more closely resembles a 50 50 split will occur.

 

Suggesting to me, that there should be an average taken to where certain sequences that deviate from the norm, or the pull of the law of large numbers should be engaged, as the number of trials increase. Maybe under 6 trials or so, it does not matter much, but as you go to 10 or 100 or 1000, the improbability of the long runs should cause one to lower the odds of those sequences that include long runs, and favor the sequences that do not.

 

Regards, TAR

Or in some way adjust the odds to include the law of large numbers.

I am thinking something like a bell curve arrangement, where short run sequences will gather around the norm, and long run sequence will be the outliers.

Edited October 18, 2016 by tar

[swansont]

No, the odds never have to be adjusted. What happens is that with more trials the expectation of certain results increases, e.g. the expectation of a run of 5 in a row. But the odds of it occurring haven't changed.

 

And you keep saying that something will not happen. Any sequence is equally likely, so you can say that about any particular sequence. But If you flip the coins, some sequence will result. Even though the a priori odds of it were quite small

[tar]

swansont,

 

well, I am talking about classes of sequences

 

That certain sequences are more normal looking than others, and in that, their odds of occurring should be, from the beginning, considered more likely, because the law of large numbers will prevent a string of 100 from happening, whereas it will not prevent HHTTTHTHHHTTHTHTTTTHHHHTHTHTHHHTTTHHTHTHTHTHTHHHHHHTHTHTHTHTHTHTHHTTTHTHTTHTTTHHHTHTHHTTHTTHTTTHTTH

from happening.

 

I understand that getting the above sequence is just as rare as getting 100 Heads, as there are enough other arrangements to each have equal probability and still have a very small chance of getting that particular sequence, but if I were to tell you I made up the string, or I flipped the string, you would not know for sure, which was the case, because the sequence is normal looking.

 

However, if I told you I just flipped HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTTTTTTTT

you would know I was lying, because the sequence laughs in the face of the law of large numbers.

 

Regards, TAR

After all, there is a REASON why the fair coin lands on either side with the same frequency.

Expecting 100 Heads in a row ignores this reason. Whatever that reason may be.

[swansont]

swansont,

 

well, I am talking about classes of sequences

 

I don't think anybody else has been, at least primarily. Expecting 50 heads vs some other number is a different application than looking at the odds of a particular sequence. Of all the possible sequences, there are more that have 50 heads than there are with any other number. Just as there are more ways of rolling a 7 with two fair dice than rolling a 10. But the odds of any specific result is the same, if the dice aren't identical (i.e. ordering matters) Rolling a 3 on the first die and a 4 on the second has the same odds as rolling a 5 and a 5. But there are 6 rolls that result in a 7, and only 3 that result in 10.

 

That certain sequences are more normal looking than others, and in that, their odds of occurring should be, from the beginning, considered more likely, because the law of large numbers will prevent a string of 100 from happening, whereas it will not prevent HHTTTHTHHHTTHTHTTTTHHHHTHTHTHHHTTTHHTHTHTHTHTHHHHHHTHTHTHTHTHTHTHHTTTHTHTTHTTTHHHTHTHHTTHTTHTTTHTTH

from happening.

 

I understand that getting the above sequence is just as rare as getting 100 Heads, as there are enough other arrangements to each have equal probability and still have a very small chance of getting that particular sequence, but if I were to tell you I made up the string, or I flipped the string, you would not know for sure, which was the case, because the sequence is normal looking.

 

However, if I told you I just flipped HHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHHTTTTTTTTTTTTTTTTTTTTTTTTTT

you would know I was lying, because the sequence laughs in the face of the law of large numbers.

 

Regards, TAR

 

If you told me you flipped HHTTTHTHHHTTHTHTTTTHHHHTHTHTHHHTTTHHTHTHTHTHTHHHHHHTHTHTHTHTHTHTHHTTTHTHTTHTTTHHHTHTHHTTHTTHTTTHTTH

then yes, I'd believe it. But you would not have predicted it ahead of time; that I would not believe. The odds of an event that has already occurred is 1, but the odds of predicting it is (1/2)^n

 

The odds of this particular sequence is the same as any other sequence of the same length. Same as the odds of a straight flush in spades being the same as any other five-card sequence, suit included. The alternate way of looking at it is if you dealt a large number of hands, you will get any particular sequence just as often as the straight flush in spades. That doesn't change because you've made the string longer. It only makes it harder to verify.

 

All you've done here is underscore a misconception that people have of randomness and probability.

 

After all, there is a REASON why the fair coin lands on either side with the same frequency.

 

Expecting 100 Heads in a row ignores this reason. Whatever that reason may be.

So is expecting HHTTTHTHHHTTHTHTTTTHHHHTHTHTHHHTTTHHTHTHTHTHTHHHHHHTHTHTHTHTHTHTHHTTTHTHTTHTTTHHHTHTHHTTHTTHTTTHTTH

[tar]

swansont,

 

I understand I can not predict that sequence and then roll it.

 

But the idea was that that sequence is made up of words, or sub sequences, or runs of one outcome, which are more likely than the run with which the 100 Heads sequence is made of.

Like the 7 being a more likely outcome than the 10. It is more likely that the sequence will be made of something less than 100 in a row, because there are only 2 sequences that have this characteristic. All the others do not have this characteristic, and it is way more likely a sequence that does not have 100 in a row will be flipped than one that has 100 in a row.

 

As there are way more ways to make sequences of small runs than sequences of large runs, the odds of having a small run sequence should be greater than having a large run sequence.

 

Regards, TAR