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### expected number of 1-runs

Posted: **May 6th, 2020, 12:10 pm**

by **AnonymousQuantus**

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.

### Re: expected number of 1-runs

Posted: **May 7th, 2020, 3:09 pm**

by **katastrofa**

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

### Re: expected number of 1-runs

Posted: **May 7th, 2020, 6:57 pm**

by **AnonymousQuantus**

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.

### Re: expected number of 1-runs

Posted: **May 10th, 2020, 2:11 pm**

by **Alan**

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.

### Re: expected number of 1-runs

Posted: **May 10th, 2020, 4:15 pm**

by **AnonymousQuantus**

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

### Re: expected number of 1-runs

Posted: **May 10th, 2020, 4:19 pm**

by **Alan**

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

### Re: expected number of 1-runs

Posted: **May 13th, 2020, 2:03 pm**

by **Alan**

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).

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 12:18 am**

by **katastrofa**

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?

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 12:07 pm**

by **Cuchulainn**

Too bad the OP seems to be MIA.

Ask Naughtius Maximus?

www.youtube.com/watch?v=kx_G2a2hL6U

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 1:32 pm**

by **Alan**

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)

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 4:52 pm**

by **katastrofa**

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

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 5:29 pm**

by **Paul**

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

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 5:49 pm**

by **Alan**

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

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 6:57 pm**

by **Cuchulainn**

Or B_Swinging_D?

### Re: expected number of 1-runs

Posted: **May 15th, 2020, 7:10 pm**

by **Alan**

Michael Lewis or Monty Python -- take your pick.