Math Is Fun Forum

grad

Member

Registered: 2009-12-06

Posts: 3

Runs in row of coins

Hi,

I spent a couple of days looking for the answer for the above problem, so I would be very grateful to anyone who can help me with this one:

Consider a row of N unbiased coins. You can choose randomly one of these coins. What is the probability that this coin belongs to a run of L heads (or tails)?

In other words I want to know the probability that a coin belongs to a run of exactly L consecutive heads (or tails) in N independent tosses of a coin.

Thank you in advance for your help...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Runs in row of coins

Hi grad;

Go here:

http://www.mathisfunforum.com/viewtopic … 46#p124546

You must be able to follow what has been done there. Although your problem is different I will use the same methods to try and solve it. Please understand that your problem , like tongzilla's problem is very difficult.

Also, in my estimation:

Consider a row of N unbiased coins. You can choose randomly one of these coins. What is the probability that this coin belongs to a run of L heads (or tails)?

In other words I want to know the probability that a coin belongs to a run of exactly L consecutive heads (or tails) in N independent tosses of a coin.

Are you aware that these two statements are not the same? It will be easier to solve the first one.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

grad

Member

Registered: 2009-12-06

Posts: 3

Re: Runs in row of coins

Maybe I didn't use the correct words but English is not my first language

What I am asking for is the answer for the first statement (it's a different probability than tongzilla's and I think that it is easier to solve).

Sorry if I confused you with the second statement...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Runs in row of coins

Hi grad;

Nothing to be sorry about, just trying to clearly define the problem before we start.

I tend to disagree that this will be easier to solve.

First of all can we relax the condition for N, in other words can you be happy with some fixed integer say N = 20 or N = 30?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

grad

Member

Registered: 2009-12-06

Posts: 3

Re: Runs in row of coins

I would like the final answer to be independent of the number but it's ok to start with a fixed number. I would prefer larger numbers if that matters because I am going to use this formula for N->∞

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: Runs in row of coins

Hi grad;

Excellent, that is all I needed to hear! That simplifies the problem somewhat. I will post when I have something.

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

mathsyperson

Moderator

Registered: 2005-06-22

Posts: 4,900

Re: Runs in row of coins

I'd say having an infinite row of coins makes the problem simpler. That way, you don't have to worry about picking a coin near the edge - any coin you pick will have L coins to its left and L coins to its right.

Here's my thinking. If you want the coin to be part of an 3-long chain, for example, then it has to be part of a string that looks like THHHT (or HTTTH).

However, it gets 3 chances to get this right. If X is the coin you select, then any of these will work:

THHXT--
-THXHT-
--TXHHT

Each of these has four coins that we require to be specific sides, so the chance of each one happening is 1/16.

Since there are three possible ways it can happen, the chance of your coin being part of a 3-chain is 3/16.

Adding the probabilities together like that is fine, because the three events are all mutually exclusive.

In general, the probability of your coin being part of an n-long chain is:

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