Jarlaxle

Premise: Flip a coin an infinite number of times then 50% of the time it will come up heads and 50% of the time it will come up tails. (I think that is correct.)

Question if after infinity half of the time the outcome will be tails, what is the maximum number of times one can flip a coin and come up heads? i.e. is there a point in probability that we have determined to be so small that we say it is impossible even though the probability is not actually zero?

Sorry if I have totally mutilated this question; if you don't understand what I am attempting to ask, I will try again. :banghead:

Shadowy Man

I'm not sure I understand your question, but in theory you can flip a coin and get heads every time. After all, there's a 50/50 chance of getting heads each time.

The odds of the next flip are unrelated to the results of the previous flips.

So, I guess there is no maximum number of times you can flip heads.

Mageth

I'm not sure I understand your question, but in theory you can flip a coin and get heads every time. After all, there's a 50/50 chance of getting heads each time.

If I understand the question right, though, he's talking about the maximum number of times (say N) that one could reasonably expect to flip a coin with it coming out heads every time. In other words, a series of flips must be considered, not just one flip.

For N=2, the probability that both flips will be heads is .25 (.5 * .5, or .5^2). For N=3, the probability is .125 (.5^3)

So, for N flips, the probability of all flips being heads is .5^N.

It's easy to see that, for some value of N, the probability will indeed become so small that one could indeed say "it is impossible even though the probability is not actually zero".

That value of N is, IMO, is somewhere between 100 and 200 flips (or perhaps even fewer). The probability of hitting heads 100 times in a row with an unbiased coin and flipper is approximately 7.9x10^-31; 200 times is approximately 6.2x10^-61.

Shadowy Man

True, eventually you will have basically a zero chance that you would have flipped that many heads.

Amazingly enough though, this would have no impact on whether or not the next flip was heads!

Mageth

That is true.

leyline

just to throw a spanner in the works....

it may depend upon how fast you can flip the coin. I mean i agree with what you guys say, but how does 6.2x10^-61 compare to the probability say of life developing on a planet?

Or a nucleus decaying?

In other words is there a probability that is too remote for the universe to allow it, irrespective of how many times that 'event' is tossed in a second? What this may mean is that the ultimately small probabilty is directly related to the ultimately high frequency (or shortest time duration) and how long that event can meaningfully survive the tossing. (After all even a coin will eventually wear out)

Another way of looking at it is to ask what probability means anyway. Does it mean a lack of information? Thus the probability that the coin will land a hundred heads in a row is dependent upon us not knowing the micro data that might accurately predict it. Knowing that knowledge reduces the probabilty to 1 or zero. Thus probabilties of between 1 and zero become measurements of partial knowledge and thus the question becomes.......

what is the smallest amount of knowledge a human being can percieve? lol

Jarlaxle

I have often heard that if a monkey randomly typed on a type writer for infinity, then it would eventually type every book that has been written. Sounded right to me in the past, but this "coin flipping" idea has made me think again.

Some sets of numbers are bigger than other sets even if both are "infinite" in size. The real numbers are uncountably big where as the whole numbers are infinite. That is, the set of real numbers is infinitly bigger than the set of whole numbers.

What I am getting at, is that even given a set that is infinite in size, there are still things not in that set. Likewise, even if the monkey typed for infinity, there could still be things that had never gotten typed.

One way to look at why the set of real numbers is bigger than the set of whole numbers is that there is no smallest number greater than zero in the real number set; no matter what number you choose, there is always a smaller one.

Question: is time like the set of real numbers in that there is no smallest unit of time or is time like the whole numbers where there is a smallest unit of time?

Dubin

I do believe time is like the real number set, seeing as how we always can and do divide our current smallest unit of time into even smaller units. If there's a smallest unit of measuring time, why not one for real numbers?

callmejay

I don't think this would apply to the monkeys, since the set of all possible books is the same size as the set of all integers. You can prove this easily by mapping each character (a-z, A-Z, 0-9, etc.) to an integer. ASCII is an example of such a mapping. The set of all integers is the smallest kind of infinity, and it includes the set of all possible books. Therefore, all books would be typed.

Eric H

Quote Mageth
It's easy to see that, for some value of N, the probability will indeed become so small that one could indeed say "it is impossible even though the probability is not actually zero".

That value of N is, IMO, is somewhere between 100 and 200 flips (or perhaps even fewer). The probability of hitting heads 100 times in a row with an unbiased coin and flipper is approximately 7.9x10^-31; 200 times is approximately 6.2x10^-61.
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I tend to agree with this view, that there are limits.

As to the monkey question it was dealt with in another thread when this experiment was actually put to the test. It was found that the monkeys became bored quickly and preferred to defecate on the typewriters and break them. From memory I think they kept pressing the same keys, and it was felt unlikely they could type more than a sentence or two by chance.


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Quote leyline: it may depend upon how fast you can flip the coin. I mean i agree with what you guys say, but how does 6.2x10^-61 compare to the probability say of life developing on a planet?
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You need very high numbers for it to have a chance, and when you are dealing with single cell life that could possibly happen because of the numbers possible. But when you talk about animals the size of a mouse there are smaller numbers to work with, less for animals the size of a horse and even less numbers for animals the size of a whale.

For evolution to happen you need a population of a species not just a solitary animal, which seems even more unlikely to my way of thinking.


peace

Eric