Math Is Fun Forum

nikov

Member

Registered: 2009-04-27

Posts: 1

Coin tossing

I toss a fair coin 50 times in a row, and write down the result on a sheet of paper (you cannot see this yet). On another sheet of paper I write whatever I want. Then I shuffle two sheets and show them to you. One of them contains 00111100000100110100000111010111101000111101011010, and the other 00000000000000000000000000000000000000000000000000. Now you have to guess which of the sheets contains the result of the coin tossing. Can you do better then randomly select a sheet (that would give you success probability of 1/2)?

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Coin tossing

I guess the first one.
Technically, the results on the sheets both have the same chance of occuring: (1/2)^50.
The first sheet has properties that are far more likely, ie, roughly even amounts of 0's and 1's and no discernable pattern, but that particular 50-digit string is no more likely to occur than the second one by coin tosses.

However, your brain is another matter.
If we flip the question around, and ask "Which sheet did you make up numbers for?", then the second one seems much more likely. Humans like patterns, and depending on your mood, there's maybe a 0.1% chance that you'd decide to write that down.
But since the first sequence is "random-looking", the chance of you writing that is closish to the chance of coin tosses producing it (so much less than that of the all-zeroes).

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Why did the vector cross the road?
It wanted to be normal.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Coin tossing

there's maybe a 0.1% chance that you'd decide to write that down.

Why is it that people feel the need to quantify everything?  It's rather unfortunate that when you don't know a quantity, humans tend to make things up instead of saying "a small amount". (Just a pet peeve, nothing more)

But you bring up a good point.  The motives of the person writing it down are suspect to suspicion.  And whenever you have psychology entering into a math problem, the typical answer is that the problem is not well-defined, at least from a mathematical point of view.

Of course, there are other reasons why this question is not well-defined.  For example, as mathsyperson says each have an equal number of chances of occurring.  However, a sequence with 20 1's and the rest 0's in it is much more likely to occur than a sequence with all zeros.  The problem lies with statistics: statistics are mathematically generated, but they can not be mathematically interpreted.

Then you can get into the order of the coin flips and distributions.  I think it would be hard for anyone to say a coin that flips 1000 heads and then 1000 tails is evenly weighted.

Because of all of this, my answer would be no, it is not possible to do better than random.  That said, if I were a betting man, I'd have to go with the obvious choice.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Coin tossing

I get your point, and my 0.1% was indeed a complete guess. I just wanted to assign some rough order of magnitude to it.
Just saying "a small amount" would have been confusing since my point was that it was actually comparatively huge.

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Why did the vector cross the road?
It wanted to be normal.