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AnonymousQuantus
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Joined: May 9th, 2012, 12:15 pm

### expected number of 1-runs

What's the expected number of 1-runs when tossing a fair coin 100 times;
where 1-run (for heads) is defined as HT or H for the last toss;
for example: THHTHTHTTH has 3 1-runs.

katastrofa
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Location: Alpha Centauri

### Re: expected number of 1-runs

The max number is 50, the min number is 0, the average is 25.

AnonymousQuantus
Topic Author
Posts: 9
Joined: May 9th, 2012, 12:15 pm

### Re: expected number of 1-runs

Can not find fault in your reasoning: (50+0)/2 = 25 and it works for n=2;
althought n=100 and we are looking for a mean not an average.

Alan
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### Re: expected number of 1-runs

Numerically, it looks like

$\mbox{mean}_{100} \approx 12.75 \pm 0.003$,

and for n draws,

$\mbox{mean}_n \sim \frac{1}{8} n$, as $n \rightarrow \infty$.

Also, makes sense that the $\mbox{mean}_n/n$ is decreasing with $n$, as the above suggests.
After all, a length-$n$ sequence ending in TH would have a 1-run counted there, but not necessarily if that sequence was continued.

Looking at Feller, he suggests looking at this type of problem as a recurrent process, but I didn't have the patience. In other words, once a 1-run occurs (say in a sequence of indefinite length), the problem of the next 1-run is (an independent) probabilistic replica of the original problem.

I'm sure there must be a nice argument for my asymptotic guess; maybe even a nice formula for finite n. I would be interested to see them.

Anyway, that's as far as I got -- I suggest Feller Vol. I for general strategy hints.

AnonymousQuantus
Topic Author
Posts: 9
Joined: May 9th, 2012, 12:15 pm

### Re: expected number of 1-runs

That's spot on. It is exactly 12.75 (and stdDev = 3.326...).
my computation indicates that

mean(n) = n/8 + 1/4 for n > 1

Alan
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### Re: expected number of 1-runs

Nice and simple. Can you post how you got it?

Alan
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### Re: expected number of 1-runs

Too bad the OP seems to be MIA.

I thought the problem was interesting.

Can anybody derive AQ's (likely correct) formula?
(I will try again at some point).

katastrofa
Posts: 10067
Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: expected number of 1-runs

Now I understand the question is about just H or just T single runs -?
The guessed formula isn't correct IMHO.
It can be solved either inductively or deductively. Who do you like more, Newton or Sherlock Holmes?

Cuchulainn
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### Re: expected number of 1-runs

Too bad the OP seems to be MIA.

"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

Alan
Posts: 10593
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: expected number of 1-runs

Now I understand the question is about just H or just T single runs -?
The guessed formula isn't correct IMHO.
It can be solved either inductively or deductively. Who do you like more, Newton or Sherlock Holmes?

Single H runs in a sequence of n coin tosses consist of:
HT...       at the beginning
.. THT … prior to the end
… TH      at the end

Looking for a formula + derivation for the mean number of single H runs in a sequence of n tosses.

The proposed formula seems likely correct, given my numerics and small n cases. If so, all we need is the derivation. Whoever provides it shall be named honorary BD (see Monty Python link)

katastrofa
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Joined: August 16th, 2007, 5:36 am
Location: Alpha Centauri

### Re: expected number of 1-runs

If you like that formula, I will rather not post the solution, which shows that it's incorrect (-:

Paul
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Joined: July 20th, 2001, 3:28 pm

### Re: expected number of 1-runs

Where’ve you been for the last two months? The more wrong the formula the more we lap it up!

Alan
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### Re: expected number of 1-runs

Come on, people -- this problem can't be that hard! Let's wrap it up so we can go back to the pandemic.

Cuchulainn
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Joined: July 16th, 2004, 7:38 am
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### Re: expected number of 1-runs

Or B_Swinging_D?
"Compatibility means deliberately repeating other people's mistakes."
David Wheeler

http://www.datasimfinancial.com
http://www.datasim.nl

Alan
Posts: 10593
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

### Re: expected number of 1-runs

Michael Lewis or Monty Python -- take your pick.