Coin Toss Question

Discussion in 'Recreational Math' started by joe14682, Oct 5, 2004.

1. joe14682Guest

Would someone please settle this argument?
My position is: 0.5^1001

Thanks!

If a fair coin turns up "heads" a 1,000 consecutive times, what is
the probability that it will turn up "heads" in the 1,001th toss?

joe14682, Oct 5, 2004

2. VirgilGuest

Depends on the coin and what you mean by "fair". A "fair" coin, in the
usual sense of "fair", has a very small but non-zero probability of
turning up heads 1000 times in a row, in which case, the probability of
one more head from the still "fair" coin is still 1/2.

Virgil, Oct 5, 2004

3. DaveGuest

The probability of getting a head on the next toss is 50% regardless of
what went before, even if that was a string of thousands of heads.

However there is a substantially different question you may be confusing
this with: which is more likely given 1001 tosses - 1001 heads, or 1000
heads then one tail, to which I'd guess the answer is the last one.

Dave, Oct 5, 2004
4. Mike TerryGuest

No, both your outcomes are still equally likely, each occuring with
probability 0.5^1001.

However there is a substantially different question you may be confusing
this with: which is more likely given 1001 tosses - 1001 heads, or 1000
heads and one tail, to which the answer is the last one.

This is because in this case, the tail could occur in any of the 1001
tosses, not just the last, so the second outcome is 1001 times more likely
(probability = 1001 * 0.5^1001)

Regards,
Mike.

Mike Terry, Oct 5, 2004
5. VirgilGuest

Of course, if 1000 tosses in a row are all heads, one might begin to
doubt the "honesty" of the coin.

And if one's doubts are well founded, 1001 heads might be more likely
that 1000 heads and one tail in the next sequence of tosses.

Virgil, Oct 5, 2004
6. OppieGuest

No one ever said life was fair. Neither are coins. I happen to know that
coin personally and I can tell you that it is neither fair nor
reasonable. It loves to do things like this just to lull you into a
false sense of security.

Trust me .. the next coin toss is almost certainly a 'tails'.

Never trust that coin. Never. It's evil ... evil I tell you!!!

Oppie, Oct 6, 2004
7. Joseph MiskoGuest

Empirically you might doubt. But just because a coin turns up 1000
consecutive heads is not proof that it is not fair. In fact, a perfectly
fair coin cannot be demanded to produce any finite number of heads or tails
that are even close. It is possible for a coin to produce a million heads
and ten tails and still be totally fair. This shows you cannot use a finite
trial to demonstrate that a coin is fair.

I wonder if there is any practical, finite method that can be used to
demonstrate that a coin is in fact fair or not.

Joe

Joseph Misko, Oct 14, 2004
8. burtGuest

I respectfully disagree. In all the history of the world from the big
bang to the big crunch, a fair coin would NEVER EVER turn up 1000
consecutive heads. I challenge you to give any remote example where
something this unlikely has ever occurred.For example, if you look at
the expansion of pi or "e" to the,say, 10**1000 places, you would
never find a sequence of 1000 consecutive even integers. The
mathematics in examples like these are theoretically interesting, but
lead to fallacious conclusions-the fallacious conclusion being that
anything this remote will actually happen sometime someplace. If a
coin has turned up 1000 consecutive heads, bet the ranch and all your
belongings that the next flip is also a head.

Burt

This shows you cannot use a finite
I respectfully disagree.

burt, Oct 15, 2004
9. Joseph MiskoGuest

Glad to hear you respectfully disagree! That's rare!

I think you are missing one thing. You are arguing based on observation and
practical experience of short-term experiments, such as everyday life
consists of. The real shocker in all this is that mathematics is TRUE!
That means, given one toss, you have half a chance to get a tail. Given two
tosses you have a quarter chance to get two tails. Three tosses means that
really and truly and practically, not "theoretically", you really do have an
eighth of a chance of getting three tails in a row. So, does that mean that
if I grabbed a fair coin right now and tossed it one thousand times, that I
would get a thousand heads. No. I agree with you that I would bet the
ranch if it did. However, such an event is possible. The minute we say it
would NEVER EVER happen, we bend the laws of nature. Statistics informs us
that it not only can happen, but does. There is nothing to reduce the odds
of three out of three tails below one-eighth. There is nothing at all,
nothing, that will reduce the odds of getting a thousand heads in a row to
zero. This event is possible and there is no way to disallow it as
possible. But the real crux of my statement lies below.
This is my point. Getting ten tails in a row does not "prove" that the coin
is biased. If it does, then at what number of tails in a row do we cross
the magic threshold and are able to declare the coin is biased? Ten? Five?
Three? One hundred? Such a test does not exist.

Two tangents before I go -- first, there will be no big crunch, unless you
mean when the rock hits you, and second, if you had started flipping a penny
at the "big bang" as they call it, you would have gotten quite a number of
strings of one-thousand heads or tails by now, depending on how fast you
flip.

Take it easy,
Joe

Joseph Misko, Oct 15, 2004
10. mensanatorGuest

What if the universe oscillates, endlessly cycling back and forth from
Big Bang to Big Crunch? How many flips can you do in one Big Bang/Crunch
cycle? 10*100? Then you would only have to wait 10**201 cycles, which
would be a doddle to infinity.
Why not? Have you checked? If the odds of 1000 in a row are 1/10**301,
what probability theorem is telling you it can't happen in 10**1000 trials?
And when you do, you'll be shocked to discover that the _coin_ was indeed
fair but the flipper was cheating. He was controlling the outcome to
sucker you into betting the ranch.

mensanator, Oct 15, 2004
11. ProginoskesGuest

Maybe not. However, there is a way to use an "unfair" coin in a way to simulate
a fair coin. What you do is to toss the "unfair" coin. The probability of
HT and TH are both equally likely. (If the probability of getting heads is p,
these probabilities are p(1-p).) If both tosses are heads or both tails,
toss it twice again, etc.
-- Christopher Heckman

Proginoskes, Oct 16, 2004
12. mensanatorGuest

Really? You do realize that to expect a single occurrence of 1000 in a row
you have to flip 10**301 times? Even if you flipped at the unimaginable rate
of 10**100 flips per second, how many would you have done since the Big Bang?

Have you ever actually worked out the math? It's fascinating, see

http://members.aol.com/mensanator666/fun/2000_bit.htm

Take out the part about parallel universes and reduce to a single coin
to get the above problem. I would say "quite a number" is something of
an overestimate.

mensanator, Oct 16, 2004
13. Joseph MiskoGuest

Yes, I have. First, to *expect* an occurrence of 1000, I think (?) you only
need to flip half that many . . . but same point! ;^D

Next, you may still be missing the point. This discussion was moving along
the lines that this would never happen. Actually, there is a possibility
greater than zero that the first thousand flips could all be tails. Not a
great possibility, but statistics doesn't allow us to say that the first
thousand flips *cannot* be tails, even from a fair coin. In fact, if we
took a googol coins and flipped as you say at the rate of about a googol per
second it wouldn't take too long. It could even happen in the first decade
or the first minute. Then again, we could still be flipping. But I am not
talking about what we can expect, I am talking about a coin that is reputed
to be fair and turning up 1000 heads in a row. It is possible, so while we
do not expect to see it, we cannot say that a fair coin is unable to produce
1000 tails and still be fair. In fact, achieving 1000 tails, or 100 or 10,
is well nigh a possibility.

Take it easy,
Joe

Joseph Misko, Oct 23, 2004
14. Bob HarrisGuest

The sun coming up. There's a 50/50 chance each morning that the sun will
stay on the other side of the earth. To the best of my knowledge is hasn't
happened yet. But it's an even bet that tomorrow morning it'll happen.
I think you'd have joint problems before completing this experiment.

Bob H

Bob Harris, Oct 24, 2004
15. mensanatorGuest

No there isn't. Probability is the ratio of successful outcomes to
the total possible outcomes. There is one possible outcome and there
is no chance that the sun will stay on the other side of the Earth.
The probability is therefore 0/1.
Because staying on the other side of the Earth is not a possible outcome.
Absolutely false. A 0/1 probability compared to a 1/1 is never an even bet.

mensanator, Oct 24, 2004
16. OppieGuest

Using past occurences to predict the future doesn't make sense in the
realms of common math. Consider the following statement:

No one in the past has ever lived more than 200 years. Therefore, we can
assume that everyone alive today will be dead in 200 years.

Is that true? How do we know that a counter-example doesn't exist? How
do we know that starting tomorrow, death shall simply never happen
again? We don't.

We have a GOOD hunch that people will continue to die. We have LOTS of
evidence to support that theory, but until it happens, you really can't say.

Toss a coin a trillion trillion times and you will almost certainly get
a billion heads in a row!

Oppie, Oct 24, 2004
17. Bob HarrisGuest

Sorry, I forgot to indicate that I was being (or trying to be) funny.

Bob H

Bob Harris, Oct 25, 2004
18. Bob HarrisGuest

Well, I just happened to see a TV show last week that said the ages of
people in the bible are accurate. A biology professor from the University
of Tennessee presented evidence that he seems to believe supports that
conclusion. He said that human DNA now has some large number of bad
mutations, and that in the early days after creation there weren't many.
That was the strongest point of his argument.

So, if you believe him, we have many counter examples to your claim, with
ages up to the 900 year range having been commonplace in the past.

Bob H

P.S. I think a more plausible explanation is that somewhere along the way
the story got mistranslated and a phrase that meant "900 lunar periods"
became one meaning "900 solar periods".

Bob Harris, Oct 25, 2004
19. OppieGuest

I do hope your response was in jest (but I really can't tell). You're
not implying that a mythical text should be used in a math argument, are
you?

If so, let's pull out the Tibetian Book of the Dead, the Hopi Stone
Tablets, and the Kami of Shinto.

Oppie, Oct 25, 2004
20. OppieGuest

Just read your other response. AHHH .. just a joke. Whew! Had me worried
for a minute.

But it would be interesting to see what the Hopi Stone Tablets DO say!

Oppie, Oct 25, 2004

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