Is pi a truly random sequence?

[cpend]

Hey, first post. Looking forward to many more!

Me and a friend were having a discussion about probability regarding lottery numbers. We talked about randomly choosing numbers and I raised the question: Is it possible to "Choose" random numbers? The fact that you are deciding which numbers to choose, does that mean they're NOT random. Or is it a paradox?

I mean, you might think that you're choosing numbers randomly but if you're thinking about which one to choose then it can't be random. Subconsciously there might be a logic pattern to your choice, unless of course you close your eyes and pick them out of a hat.


Anyone have any insight into this?

Cheers

 

[DaveC426913]

I would certainly agree. Chosen numbers are surely not very random.

Someone, somewhere documented the distribution of subjects choosing a number between one and ten (where you want to stump the tester). The numbers 3 and 7 were chosen more often than others. The logic goes that these numbers are psychologically more "hidden" than other numbers (such as 1 or 5).



Though, as I'm sure you're aware, you *don't* have to choose random numbers. The randomization is inherent in the results, not the picks. You could pick the most *un*random set of numbers you can imagine.

I encountered this when discussing lottery numbers with my brother. I could not convince him that picking 1,2,3,4,5,6,7 was just as likely to win as any other string of 7 digits. He saw this as an extremely unlikely series of numbers to win.

 

[HallsofIvy]

I think the problem is with the phrase "random number" itself- which ought to be outlawed! A specific number is not random so there really is no such thing as a "random number". What is meant is a "randomly chosen" number. In fact, your title to this thread "Choosing a number randomly" says precisely that!

If I flip a coin, and for heads, choose "1", for tails, choose "2", then obviously neither 1 nor 2 is a "random number" (just imagine going up to your math teacher and asking "is 2 a random number?") but my method of choosing them is random. I might also speak of finding a "random sequence" of numbers that way- although after I have chosen one: say, "011100110101"- it certainly is not random! It was rather, "randomly chosen".

 

[cpend]

I think the problem is with the phrase "random number" itself- which ought to be outlawed! A specific number is not random so there really is no such thing as a "random number". What is meant is a "randomly chosen" number. In fact, your title to this thread "Choosing a number randomly" says precisely that!

Yes you're right, I'll edit my post accordingly.

I encountered this when discussing lottery numbers with my brother. I could not convince him that picking 1,2,3,4,5,6,7 was just as likely to win as any other string of 7 digits. He saw this as an extremely unlikely series of numbers to win.

I had the EXACT same conversatin just yesterday.

Someone, somewhere documented the distribution of subjects choosing a number between one and ten (where you want to stump the tester). The numbers 3 and 7 were chosen more often than others. The logic goes that these numbers are psychologically more "hidden" than other numbers (such as 1 or 5).

That's the kind of results I would've expected as well. Thanks guys

 

[Jimmy Snyder]

I encountered this when discussing lottery numbers with my brother. I could not convince him that picking 1,2,3,4,5,6,7 was just as likely to win as any other string of 7 digits. He saw this as an extremely unlikely series of numbers to win.

Is there a mathematical, or psychological concept of 'random appearance'?

While this string of numbers is no more and no less likely to win than any other sequence, I think it is more likely that the pot will be shared with others if it does win. If you had a list of bettable sequences and the frequency with which they are bet, I expect that, in general, the least frequently chosen would have some 'random appearance' to them that the most frequently chosen do not have.

 

[robert Ihnot]

What you might do is find a published sequence of numbers like pi to ten thousand decimals, since it is considered a random sequence, that might do. Or you could throw dice, etc.

 

[robert Ihnot]

jimmysnider: I expect that, in general, the least frequently chosen would have some 'random appearance' to them that the most frequently chosen do not have.

Years ago in the Ohio Lottery, I heard that many people pick numbers based on birthdays, so that numbers greater 31 are, possibly, less likely to be chosen. I might suspect that primes over 31 are seen as "ackward numbers" not related to much, and not likely to be chosen. Any sequence like 1,2,3,4,5,6...would probably get players because it is so easy to think of.

Furthermore, I would avoid numbers, that from the standpoint of the ticket are on corners or diagonials as these are easy to pick. "Unpopular numbers" will not win anymore often than others, but may make it less likely to share a prize.

If you are really into this, get a list of winning numbers in the past and see how many of these cases had multiple winners, etc.

 

[HallsofIvy]

What you might do is find a published sequence of numbers like pi to ten thousand decimals, since it is considered a random sequence, that might do. Or you could throw dice, etc.

Considered a "random sequence" by whom? I know that pi is believed to be a "normal" number but that has never been proven. Also I would argue that it is completely incorrect to say that a number whose decimal expansion is completely determinate gives a random sequence!

 

[robert Ihnot]

HallsofIvy: Considered a "random sequence" by whom?

I have a source headed by: Pi seems a good random number generator – but not always the best. http://news.uns.purdue.edu/html4ever/2005/050426.Fischbach.pi.html

Article goes on to add: they have found that while sequences of digits from pi are indeed an acceptable source of randomness...pi's digit string does not always produce randomness as effectively as manufactured generators do.

 

[HallsofIvy]

That's not quite what you said!

Saying "Pi seems a good random number generator " or "sequences of digits from pi are indeed an acceptable source of randomness..." is not at all the same as saying pi is " considered a random sequence".

 

[robert Ihnot]

HallsofIvy: Saying "Pi seems a good random number generator " or "sequences of digits from pi are indeed an acceptable source of randomness..." is not at all the same as saying pi is " considered a random sequence".

Here is agreement with your statement: http://www.everything2.com/index.pl?node_id=840430

A common mistake is to assert that the digits of pi are random. This is wrong by definition. Pi's digits are fixed and may be computed formulaically and hence are instantly distinguishable from a random sequence. The question people really want to know is whether or not the digits of pi are distinguishable by inspection in any obvious way from a random sequence, such as whether there are biases or recurring patterns in the digits. None have yet been found.
The most common formal term called into play is whether or not pi is normal. This conjecture is widely believed to be true and has held true for the billions of digits found so far. However no component of this conjecture has been formally proven for any base.

But by what I was saying, What you might do is find a published sequence of numbers like pi to ten thousand decimals, since it is considered a random sequence, was meant to say that a ten thousand decimal segiment of pi could be used as a random sequence. (In any case, it seems to have been used for that kind of purpose.)

I remember reading that a code could be devised by having both communicating individuals resort to Bible passages going letter by letter to change the cypher at every point. Then it being argued that a decoder might be able to reverse the process and detect the passages being employed, so that only a random sequence known only to two individuals would produce an unbreakable code. Maybe the same reasoning could be applied to break a code from 10,000 digits of pi, but it would be much harder to detect than Bible passages. Of course, you would not start at the beginning with 31415... I have another quote below:

Example: If the CSPRNG being considered produces output by computing some function of the next digit of pi, it may well be random, as pi appears to be a random sequence. However, this does not satisfy the next-bit test, and thus is not cryptographically secure. There exists an algorithm that will predict the next bit.http://en.wikipedia.org/wiki/Cryptographically_secure_pseudo-random_number_generator