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andy wrote:And yet if it is possible to calculate the X from question 1, there must be a method to find 'X2' which tells us how many times we have to toss to be 100% sure that the coin has flipped to the same side a staggering 20 billion consecutive times; ...

There is a method, but it won't give a specific X (or "N," which is more usual for integers). The probability function P(N) is a function of N were N is the number of flips. P(N) will be less than 1 (i.e. less than 100%) for any number of flips that is finite. For some very large N the probability will be 99.99999999999999999999999999999999999999999999999999999999%. For some much larger N, there will be a billion 9s, or any number of 9s you can name, but P(N) will not be 100% for any finite N.

But when I said "given sufficient time," that means there is no limit on time or the number of flips--you keep flipping until it happens.

In mathematical notation, you would integrate the probability function over the number of flips N, where N goes from N = 0 to N = INFINITY.

The integral of the probability function integrated from N = 0 to N = INFINITY will be 1 (100%).

There is no such thing as failure in this exercise because there is no giving up until you succeed (i.e., you don't stop at any finite number). As long as you haven't flipped 20 billion heads in a row, that just means you keep flipping.

And it will happen at a finite number N (even thought the calculated P(N) will be less than 1). It could happen at P(N) = 50%, or P(N) = 99%, etc. It can happen at P(N) = any number; it could even happen on the first try. Infinity can't be the destination because infinity isn't actually a number.

In other words, you will, with 100% certainty, get 20 billion heads in a row because you won't stop until you do.

Just as we know INFINITY isn't a number, we also know that NEVER is not a number either.

So unless you would be able to prove that the universe is infinte in its materiality, statistics could never guarantee us with 100% certainty that there is at least one other planet with life on it.

Correct?

Correct, but I'm not sure what you are getting at.

I think we started with this:

pinhedz wrote:I think that most scientists believe there are other planets like ours.

Anything wrong with that?

I already answered that question:

Andy wrote:While I think I'm basically agreeing with most of what you say, I would like the use a demonstration of why I find the 'flipping coins'-analogy inadequate to point out where I think I'm taking a different perspective.

So nothing wrong with that.

I'm not entirely sure how you manage to go from:

pinhedz wrote:In fact, if I watched a coin land heads up 20 times in a row, I would begin to question my own sanity and senses.

to

pinhedz wrote:In other words, you will, with 100% certainty, get 20 billion heads in a row because you won't stop until you do.

in the space of a single page. But that's irrelevant to the matter at hand.

P.S.: I do know 'infinity' is part of the answer, yes. Still: going from questioning your own sanity if a result is produced 20 times to assuring that it must be produced 20 billion times - you gotta admit it sounds a bit fishy.

Andy wrote:P.S.: I do know 'infinity' is part of the answer, yes. Still: going from questioning your own sanity if a result is produced 20 times to assuring that it must be produced 20 billion times - you gotta admit it sounds a bit fishy.

I don't know why that's fishy--after all, the number of flips I could execute in my lifetime is infinitesimal. To say 20 billion heads is inevitable is not the same as saying we would expect to ever see it. Nonetheless, given unlimited time, heads 20 billion times is just as inevitable as heads one time.

Yes, I really said that--heads 20 billion times is just as inevitable as heads one time.

If you flip a coin 20 times, are you 100% certain you'll get heads even once? How about 200 times? If you flip a coin 20 billion times, are you 100% certain you'll get heads at least once? The answer is No for any finite number, but yes if the limit is infinity--whether we're talking about heads one time or 20 billion times.

As for my being crazy, there is some probability of that (I'd say it's a low probability) and there is also a probability (also low) of 20 heads in a row. On balance, I think the probability that I am crazy is higher than the probability of 20 heads in a row. That is why I would question my own senses.

Unless of course I actually kept flipping until I could reasonably expect to get 20 heads in a row, in which case I would certainly (with 100% certainty) have to be crazy.

From the ZD Net UK site:

That Random Coin Toss?

By timdaily , 13 December, 2009


One of my all-time favorite scenes in a play and movie, is the scene in Tom Stoppard's Rosencrantz & Guildenstern Are Dead where every coin toss comes up heads, leading to a bit of a philosophical discussion on probability. Of course, the randomness of the coin toss is the quintessential example of a random event and is used regularly for a variety of situations in which randomness is required, let alone expected. Except... it turns out the common wisdom may be wrong. Paul Kedrosky has the news of a test that showed that if you ask people to try flip a coin and get more heads than tails, they will, and not by a small margin either. In the test, 13 people were asked to flip a coin 300 times, trying to get as many heads as possible. All 13 participants got more heads than tails. Seven out of the thirteen had statistically significant margins of heads over tails (meaning almost certainly not a matter of chance). The highest was one individual had 68% of the coin flips land heads. In other words, a coin toss isn't particularly random

Berger & Wyse.

http://www.youtube.com/watch?v=ZPvau0RBYPk&feature=related
Waiting For the UFO's- Graham Parker & The Rumour.

Stephen Collins

Stephen Collins