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        riddles >> cs >> Biasing a fair coin
        
(Message started by: birbal on Apr 13^th, 2011, 12:13pm)

Title: Biasing a fair coin
Post by birbal on Apr 13^th, 2011, 12:13pm

Given a function for a fair coin, write a function for a biased coin that returns heads 1/n times (n is a param).

Title: Re: Biasing a fair coin
Post by towr on Apr 13^th, 2011, 1:16pm

[hide]Create a random number by concatenating random bits; r_i = 1 if i^th throw is tails, 0 otherwise..
If SUM_i=1..m r_i/2ⁱ >= 1/n return tails, if 1/2^m + SUM_i=1..m r_i/2ⁱ <= 1/n return heads, and otherwise throw the coin to determine r_m+1 [/hide]

Title: Re: Biasing a fair coin
Post by birbal on Apr 13^th, 2011, 10:48pm

I was thinking a bit different solution. flip the coin k times ( where k is the minimum number of bits required to represent n in binary form). if i^th flip is head, make i^th bit 1, 0 otherwise. Let the new number formed is x, then
if(x == n-1)
return head;
if(x >= n)
repeat the process;
return tail;

Title: Re: Biasing a fair coin
Post by Grimbal on Apr 14^th, 2011, 7:43am

Here is a solution I believe is equivalent to towr's solution, but more computer-friendly.

// return true with probability 1/n
boolean flip(int n)
{
// random is random in range 0..(range-1)
int random=0, range=1;
while( true ){
while( range<n ){
  random = random*2 + fairFlip()?1:0;
  range *= 2;
}
if( random<n ){
  return random==0;
} else {
  random -= n;
  range -= n;
}
}
}

Title: Re: Biasing a fair coin
Post by birbal on Apr 14^th, 2011, 9:05am

@Grimbal - Its almost same as mine solution except for the part that when random >= n, i was discarding and generating random again. :D

Title: Re: Biasing a fair coin
Post by Grimbal on Apr 14^th, 2011, 9:34am

Yes. It is just a bit more stingy with fair flips.

I wonder: If you do:
while( true ){
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
}

you get a 'H' with probability 1/3. Is there a way to extend that to any 1/n? (i.e. design a sequence of flips that you repeat and gives the wanted probability to get H)?

Title: Re: Biasing a fair coin
Post by towr on Apr 14^th, 2011, 11:45am

Probably
Let's see; 0.101010101010 = 2/3
How about just taking the binary expansion of the fraction?

12/13 = 0.111011000100
How does:
while( true ){
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';
}
look for 1/13 chance of H?

(Actually, we could just invert it; same diff.)

Title: Re: Biasing a fair coin
Post by krishnav on Dec 29^th, 2011, 11:04am

on 04/14/11 at 07:43:57, Grimbal wrote:

if random is not < n the first time, then this code will return false. I don't see random going back to 0 again.
am I missing somthing?

Title: Re: Biasing a fair coin
Post by towr on Dec 29^th, 2011, 12:47pm

Random returns to 0 whenever random=n

For example take n=3, and the first two times fairFlip() return 1, then random=3 at the if statement, 3 get's subtracted. Then if the next round the first two times fairFlip() returns zero, then we get at the if statement with random=0 and return true.
In general for n=3 we have (11)*00=>true, and (11)*01=>false and (11)*10=>false

Title: Re: Biasing a fair coin
Post by birbal on Dec 29^th, 2011, 11:36pm

on 04/14/11 at 11:45:04, towr wrote:

Will it work for the non recurring fractions ?
for example, if i want 'H' with 1/8 probability then function should be
1/8 = 0.001
while( true ){
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
}
But here, probability of 'H' is 1/7 as a sequence of 000 would mean repeat in this case. Am i missing something ?

Title: Re: Biasing a fair coin
Post by towr on Dec 30^th, 2011, 8:46am

There's always a repeating part for a fraction for 1/8 you have repeating 0s after 0.001 (in binary). The repeating part is what is reflected in the while. If you have a non-repeated part it can be handled before the loop.

Title: Re: Biasing a fair coin
Post by krishnav on Dec 30^th, 2011, 10:04am

on 12/29/11 at 12:47:50, towr wrote:

but lets say n=6; and say random = 7 and range is 8, then the new random and range are 1 & 2 respectively.

now if the new random <6 it will return false, because we know it isn't 0.

does this still gives 1/6 probability of returning true?

may be it does, because probability of hitting n is 1/n and when it hits n we go back to 0.

how is this different from just resetting to 0 if random >= n always?

Title: Re: Biasing a fair coin
Post by towr on Dec 30^th, 2011, 11:55am

It's different because you use the bits more efficiently. It gives the same answer, but for slightly less work.

It's like if you had a die and want to simulate a coin, you could say 1=head, 2=tails, 3,4,5,6=repeat, or you could do odd=heads, even=tails, you'll get the same answer with the same probability, but in the latter case you don't waste time. You don't save quite as much time here, but you save some.

for n=6
000 = true
001 = false
010 = false
011 = false
100 = false
101 = false
110 => 0
000 => true
001 => false
010 => false
011 => false
111 => 1
100 => false
101 => false
110 => 0
111 => 1
etc

In this case you save 1 bit each round, because you reuse what's left after the previous round. So you only need two instead of three flips each additional round.
So instead of an average of 4 bits, you need an average of 3 2/3 to get the answer (if I calculated that correctly), that's 8 1/3 % faster.

Title: Re: Biasing a fair coin
Post by krishnav on Dec 31^st, 2011, 10:56am

consider this:
n = 9; so range is 16; if we generate random=10 the first time, we make random=1 and range = 7; so in one more flip range will be more than n, but random is guaranteed to be < n and isn't 0, so we return false. this means if we get 10, 11 or 12 the first time, we'll return false in the next round no matter what. isn't it?

Title: Re: Biasing a fair coin
Post by towr on Jan 1^st, 2012, 6:11am

Yes, but if you generate 9, 13 or 15 the first time round, you might return true.

0 0000 T
1 0001 F
2 0010 F
3 0011 F
4 0100 F
5 0101 F
6 0110 F
7 0111 F
8 1000 F

9 1001 -> 000
(0) 0000 T
(1) 0001 F
10 1010 -> 001
(2) 0010 F
(3) 0011 F
11 1011 -> 010
(4) 0100 F
(5) 0101 F
12 1100 -> 011
(6) 0110 F
(7) 0111 F
13 1101 -> 100
(8) 1000 F

(9) 1001 -> 000
14 1110 -> 101
(10) 1010 -> 001
(11) 1011 -> 010
15 1111 -> 110
(12) 1100 -> 011
(13) 1101 -> 100
etc

For each round, you have a 1 in 9 chance of returning true, and otherwise you go on to the next round.

Title: Re: Biasing a fair coin
Post by Grimbal on Jan 2^nd, 2012, 5:08am

on 12/31/11 at 10:56:28, krishnav wrote:

That's right, in some cases we don't actually need the last fairFlip.

Indeed, with n=9, if the first 2 flips return false then true then the random value will be in range 4..7 and the flip function will return false regardless of the 3rd and 4th fairFlip.

What happens is that I used a method designed to generate a number in range 0..n-1 and then test if it is 0 to return true with probability 1/n. We can use fairFlip more sparingly using a method like in replies #5 and #6.

Title: Re: Biasing a fair coin
Post by Grimbal on Jan 2^nd, 2012, 5:39am

Here is a function that returns true with probability p/q, given a fairFlip function that returns true and false with equal probability.

boolean flip(p,q) {
if( p>=q ) return true; // for safety
while( true ){
if( p<=0 ) return false;
if( 2*p<q ){
if( fairFlip() ) return false;
p = 2*p;
} else {
if( fairFlip() ) return true;
p = 2*p - q;
}
}
}

which can be rewritten:

boolean flip(p,q) {
if( p>=q ) return true;
while( p>0 ){
if( fairFlip() ) return 2*p>=q;
p = (2*p)%q;
}
return false;
}

Title: Re: Biasing a fair coin
Post by ashmish2 on Mar 22^nd, 2012, 3:33am

int currentloc = 0;

BiasFlip(int n)
//Circular array with all value true (tail) and 1 false
(head)

bool arr[n];
for i = to arr.length()
arr[i]=true;
arr[rand()%n]= false;

for each Fairflip() == Head
currentloc+=2
return arr[currentloc];
for each Fairflip() == Tail
currentloc--
return arr[currentloc]

Title: Re: Biasing a fair coin
Post by titan on Oct 18^th, 2013, 4:14am

I have nothing new to say. Want to confirm if the following strategy proposed by towr can be regarded as correct and can be used:-
Probably
Let's see; 0.101010101010 = 2/3
How about just taking the binary expansion of the fraction?

Quote:

12/13 = 0.111011000100
How does:
while( true ){
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'T';
if( fairFlip() ) return 'H';
if( fairFlip() ) return 'H';