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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

**A list of 1000 zeroes is created called M. A random number is picked from 1 to 1000. Each time one of these is picked, the zero in the list ( at that index ) is changed to a 1. For instance if the random number is 276 then M[276]=1. What is the average number of random numbers that must be picked to have 200 ones in the list?**

**This has an analytic solution given in the other thread.**

**http://www.mathisfunforum.com/viewtopic … 45#p227545**

This thread has been created to list the various programming solutions to the problem. Please only programming here. All analytical answers can go in the other thread.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

To check our analytic solution, we may try a simulation:

Here is an example of doing it in python

```
from random import randint
zeroes = [0]*1000
tries = 10000
hit = 0
for i in range(tries):
while sum(zeroes) < 200:
zeroes[randint(0, 999)] = 1
hit += 1
zeroes = [0]*1000
print hit / float(tries)
```

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

This idea uses a different approach. Instead of finding the number of picks it uses the analytical answer and then uses a simulation to test if it is correct.

```
fubar[l_]:=Module[{h1,g1},
g1=l;
h1=RandomInteger[{1,Length[l]}];
g1[[h1]]=1;
g1]
```

Now run this:

`Table[s=Table[0,{1000}];Count[Table[s=fubar[s],{223}]//Last,1],{100}]//Mean//N`

The output is 200.03.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**gAr****Member**- Registered: 2011-01-09
- Posts: 3,479

In R, we may speed up the simulation using the sample function:

```
sa <- c(1:1000)*0
trials <- 10000
hit <- 0
for (i in 1:trials)
{
sa[sample(1:1000, 200, replace = TRUE)] <- 1
hit <- hit + 200
while (sum(sa) < 200)
{
sa[sample(1:1000,1)] <- 1
hit <- hit + 1
}
sa <- c(1:1000)*0
}
print (hit / trials)
```

"Believe nothing, no matter where you read it, or who said it, no matter if I have said it, unless it agrees with your own reason and your own common sense" - Buddha?

"Data! Data! Data!" he cried impatiently. "I can't make bricks without clay."

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**phrontister****Real Member**- From: The Land of Tomorrow
- Registered: 2009-07-12
- Posts: 3,848

Hi Bobby,

The Olympic Games have got my attention for now but I thought I'd post this LB attempt anyway. It's a bit scratchy, but it works...so I'm happy enough.

Some better coder would be able to improve its efficiency and speed (the 10000 cycles took just under an hour!). Compared with yours and gAr's solutions, mine seems to be stuck way back in the Dark Ages!

Results: My test run of 10000 cycles yielded an average of 223 picks per cycle.

Got some long nights of TV viewing ahead!

```
FOR cycles = 1 TO 10000
M$ = ""
FOR zeroes = 1 TO 1000
M$ = M$ + "0"
NEXT zeroes
'initial 199 picks
FOR picks = 1 TO 199
random = INT(RND(1) * 1000) + 1
M$ = LEFT$(M$,random - 1) + "1" + RIGHT$(M$,1000 - random)
NEXT picks
picks = picks - 1
[morepicks]
FOR count = 1 TO 1000
sum = sum + VAL(MID$(M$,count,1))
NEXT count
IF sum = 200 THEN GOTO [nextcycle]
random = INT(RND(1) * 1000) + 1
M$ = LEFT$(M$,random - 1) + "1" + RIGHT$(M$,1000 - random)
sum = 0
picks = picks + 1
GOTO [morepicks]
[nextcycle]
totalpicks = totalpicks + picks
NEXT cycles
cycles = cycles - 1
'results
PRINT "Cycles: ";cycles
PRINT "Total picks: ";totalpicks
PRINT "Average picks per cycle: ";totalpicks / cycles
END
```

"The good news about computers is that they do what you tell them to do. The bad news is that they do what you tell them to do." - Ted Nelson

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi phrontister;

It got the right answer! Efficiency takes second place to that.

Very good work.

**In mathematics, you don't understand things. You just get used to them.Of course that result can be rigorously obtained, but who cares?Combinatorics is Algebra and Algebra is Combinatorics.**

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Hi

I get the answer of 2.23058399999999E+002:approx 223.0584 with:

```
program Expectation;
type
arr=array[1..1000] of boolean;
function count(a:arr):integer;
var
i,cnt:integer;
begin
cnt:=0;
for i:=1 to 1000 do
cnt:=cnt+ord(a[i]);
count:=cnt;
end;
var
a:arr;
i,j,number:integer;
expect:real;
begin
randomize;
expect:=0;
for i:=1 to 10000 do
begin
for j:=1 to 1000 do
a[j]:=false;
number:=0;
while count(a)<200 do
begin
a[random(1000)+1]:=true;
number:=number+1;
end;
expect:=expect+number;
end;
writeln(expect/10000);
end.
```

*Last edited by anonimnystefy (2012-07-29 03:30:03)*

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

That is very good!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

What is the actual number, again?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

And in the decimal form?

Here lies the reader who will never open this book. He is forever dead.

Taking a new step, uttering a new word, is what people fear most. ― Fyodor Dostoyevsky, Crime and Punishment

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Hi;

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

I ran the simulation once again and got the same number as before.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

The exact same number? Then you have a bug in the code.

See you in a bit. Chore time!

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Changed my code a little, but now it always gives output less then, but close to the actual answer. Haven't gotten a number over 223 with the new code.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

What is the output of a couple of runs?

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

One was something aroung 222.8 and another one was around 222.9. Let me get the actual figures.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

That seems reasonable. Over many runs you will average out to around 223.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Here is what I got now:

```
2.23019100000000E+002
2.23065900000000E+002
2.23091500000000E+002
2.23027800000000E+002
2.23021100000000E+002
2.22971200000000E+002
2.23048500000000E+002
2.23036800000000E+002
2.22945700000000E+002
2.23067400000000E+002
2.23025500000000E+002
2.23024000000000E+002
```

*Last edited by anonimnystefy (2012-07-29 08:04:28)*

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

That is about right.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

I will add a few more values in a minute.

Here lies the reader who will never open this book. He is forever dead.

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

The answer is a little more than 223, I think.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Did you see the edited post 19?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
- Posts: 87,238

Yes, it looks fine to me. The sd is very small on this problem.

Of course that result can be rigorously obtained, but who cares?

Combinatorics is Algebra and Algebra is Combinatorics.

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**anonimnystefy****Real Member**- From: The Foundation
- Registered: 2011-05-23
- Posts: 15,544

Standard deviation?

Here lies the reader who will never open this book. He is forever dead.

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